Complex multiplication for elliptic curves#

This module implements the functions

hilbert_class_polynomial
cm_j_invariants
cm_orders
discriminants_with_bounded_class_number
cm_j_invariants_and_orders
largest_fundamental_disc_with_class_number

AUTHORS:

Robert Bradshaw
John Cremona
William Stein

sage.schemes.elliptic_curves.cm.cm_j_invariants(proof=None)#

Return a list of all CM \(j\)-invariants in the field \(K\).

INPUT:

K – a number field
proof – (default: proof.number_field())

OUTPUT:

(list) – A list of CM \(j\)-invariants in the field \(K\).

EXAMPLES:

sage: cm_j_invariants(QQ)
[-262537412640768000, -147197952000, -884736000, -12288000, -884736, -32768, -3375, 0, 1728, 8000, 54000, 287496, 16581375]

Over imaginary quadratic fields there are no more than over \(QQ\):

sage: cm_j_invariants(QuadraticField(-1, 'i'))
[-262537412640768000, -147197952000, -884736000, -12288000, -884736, -32768, -3375, 0, 1728, 8000, 54000, 287496, 16581375]

Over real quadratic fields there may be more, for example:

sage: len(cm_j_invariants(QuadraticField(5, 'a')))
31

Over number fields K of many higher degrees this also works:

sage: K.<a> = NumberField(x^3 - 2)
sage: cm_j_invariants(K)
[-262537412640768000, -147197952000, -884736000,
 -884736, -32768, 8000, -3375, 16581375, 1728, 287496, 0,
 54000, -12288000,
 31710790944000*a^2 + 39953093016000*a + 50337742902000]
sage: K.<a> = NumberField(x^4 - 2)
sage: len(cm_j_invariants(K))
23

sage.schemes.elliptic_curves.cm.cm_j_invariants_and_orders(proof=None)#

Return a list of all CM \(j\)-invariants in the field \(K\), together with the associated orders.

INPUT:

K – a number field
proof – (default: proof.number_field())

OUTPUT:

(list) A list of 3-tuples \((D,f,j)\) where \(j\) is a CM \(j\)-invariant in \(K\) with quadratic fundamental discriminant \(D\) and conductor \(f\).

EXAMPLES:

sage: cm_j_invariants_and_orders(QQ)
[(-3, 3, -12288000), (-3, 2, 54000), (-3, 1, 0), (-4, 2, 287496), (-4, 1, 1728), (-7, 2, 16581375), (-7, 1, -3375), (-8, 1, 8000), (-11, 1, -32768), (-19, 1, -884736), (-43, 1, -884736000), (-67, 1, -147197952000), (-163, 1, -262537412640768000)]

Over an imaginary quadratic field there are no more than over \(QQ\):

sage: cm_j_invariants_and_orders(QuadraticField(-1, 'i'))
[(-163, 1, -262537412640768000), (-67, 1, -147197952000),
 (-43, 1, -884736000), (-19, 1, -884736), (-11, 1, -32768),
 (-8, 1, 8000), (-7, 1, -3375), (-7, 2, 16581375), (-4, 1, 1728),
 (-4, 2, 287496), (-3, 1, 0), (-3, 2, 54000), (-3, 3, -12288000)]

Over real quadratic fields there may be more:

sage: v = cm_j_invariants_and_orders(QuadraticField(5,'a')); len(v)
31
sage: [(D, f) for D, f, j in v if j not in QQ]
[(-235, 1), (-235, 1), (-115, 1), (-115, 1), (-40, 1), (-40, 1),
 (-35, 1), (-35, 1), (-20, 1), (-20, 1), (-15, 1), (-15, 1), (-15, 2),
 (-15, 2), (-4, 5), (-4, 5), (-3, 5), (-3, 5)]

Over number fields K of many higher degrees this also works:

sage: K.<a> = NumberField(x^3 - 2)
sage: cm_j_invariants_and_orders(K)
[(-163, 1, -262537412640768000), (-67, 1, -147197952000),
 (-43, 1, -884736000), (-19, 1, -884736), (-11, 1, -32768),
 (-8, 1, 8000), (-7, 1, -3375), (-7, 2, 16581375), (-4, 1, 1728),
 (-4, 2, 287496), (-3, 1, 0), (-3, 2, 54000), (-3, 3, -12288000),
 (-3, 6, 31710790944000*a^2 + 39953093016000*a + 50337742902000)]

sage.schemes.elliptic_curves.cm.cm_orders(proof=None)#

Return a list of all pairs \((D,f)\) where there is a CM order of discriminant \(D f^2\) with class number h, with \(D\) a fundamental discriminant.

INPUT:

\(h\) – positive integer
proof – (default: proof.number_field())

OUTPUT:

list of 2-tuples \((D,f)\)

EXAMPLES:

sage: cm_orders(0)
[]
sage: v = cm_orders(1); v
[(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1), (-67, 1), (-163, 1)]
sage: type(v[0][0]), type(v[0][1])
(<... 'sage.rings.integer.Integer'>, <... 'sage.rings.integer.Integer'>)
sage: v = cm_orders(2); v
 [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1), (-51, 1), (-52, 1), (-88, 1), (-91, 1), (-115, 1), (-123, 1), (-148, 1), (-187, 1), (-232, 1), (-235, 1), (-267, 1), (-403, 1), (-427, 1)]
sage: len(v)
29
sage: set([hilbert_class_polynomial(D*f^2).degree() for D,f in v])
{2}

Any degree up to 100 is implemented, but may be prohibitively slow:

sage: cm_orders(3)
[(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2), (-59, 1), (-67, 2), (-83, 1), (-107, 1), (-139, 1), (-163, 2), (-211, 1), (-283, 1), (-307, 1), (-331, 1), (-379, 1), (-499, 1), (-547, 1), (-643, 1), (-883, 1), (-907, 1)]
sage: len(cm_orders(4))
84

sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(B=None, proof=None)#

Return dictionary with keys class numbers \(h\le hmax\) and values the list of all pairs \((D, f)\), with \(D<0\) a fundamental discriminant such that \(Df^2\) has class number \(h\). If the optional bound \(B\) is given, return only those pairs with fundamental \(|D| \le B\), though \(f\) can still be arbitrarily large.

INPUT:

hmax – integer
\(B\) – integer or None; if None returns all pairs
proof – this code calls the PARI function pari:qfbclassno, so it could give wrong answers when proof``==``False. The default is whatever proof.number_field() is. If proof==False and \(B\) is None, at least the number of discriminants is correct, since it is double checked with Watkins’s table.

OUTPUT:

dictionary

In case \(B\) is not given, we use Mark Watkins’s: “Class numbers of imaginary quadratic fields” to compute a \(B\) that captures all \(h\) up to \(hmax\) (only available for \(hmax\le100\)).

EXAMPLES:

sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(3)
sage: sorted(v)
[1, 2, 3]
sage: v[1]
[(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1), (-67, 1), (-163, 1)]
sage: v[2]
[(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1), (-51, 1), (-52, 1), (-88, 1), (-91, 1), (-115, 1), (-123, 1), (-148, 1), (-187, 1), (-232, 1), (-235, 1), (-267, 1), (-403, 1), (-427, 1)]
sage: v[3]
[(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2), (-59, 1), (-67, 2), (-83, 1), (-107, 1), (-139, 1), (-163, 2), (-211, 1), (-283, 1), (-307, 1), (-331, 1), (-379, 1), (-499, 1), (-547, 1), (-643, 1), (-883, 1), (-907, 1)]
sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(8, proof=False)
sage: sorted(len(v[h]) for h in v)
[13, 25, 29, 29, 38, 84, 101, 208]

Find all class numbers for discriminant up to 50:

sage: sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(hmax=5, B=50)
{1: [(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1)], 2: [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1)], 3: [(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2)], 4: [(-3, 13), (-3, 11), (-3, 8), (-4, 10), (-4, 8), (-4, 7), (-4, 6), (-7, 8), (-7, 6), (-7, 3), (-8, 6), (-8, 4), (-11, 5), (-15, 4), (-19, 5), (-19, 3), (-20, 3), (-20, 2), (-24, 2), (-35, 3), (-39, 2), (-39, 1), (-40, 2), (-43, 3)], 5: [(-47, 2), (-47, 1)]}

sage.schemes.elliptic_curves.cm.hilbert_class_polynomial(algorithm=None)#

Return the Hilbert class polynomial for discriminant \(D\).

INPUT:

D (int) – a negative integer congruent to 0 or 1 modulo 4.
algorithm (string, default None).

OUTPUT:

(integer polynomial) The Hilbert class polynomial for the discriminant \(D\).

ALGORITHM:

If algorithm = “arb” (default): Use Arb’s implementation which uses complex interval arithmetic.
If algorithm = “sage”: Use complex approximations to the roots.
If algorithm = “magma”: Call the appropriate Magma function (if available).

AUTHORS:

Sage implementation originally by Eduardo Ocampo Alvarez and AndreyTimofeev
Sage implementation corrected by John Cremona (using corrected precision bounds from Andreas Enge)
Magma implementation by David Kohel

EXAMPLES:

sage: hilbert_class_polynomial(-4)
x - 1728
sage: hilbert_class_polynomial(-7)
x + 3375
sage: hilbert_class_polynomial(-23)
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
sage: hilbert_class_polynomial(-37*4)
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-37*4, algorithm="magma") # optional - magma
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-163)
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="sage")
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="magma") # optional - magma
x + 262537412640768000

sage.schemes.elliptic_curves.cm.is_cm_j_invariant(method='new')#

Return whether or not this is a CM \(j\)-invariant.

INPUT:

j – an element of a number field \(K\)

OUTPUT:

A pair (bool, (d,f)) which is either (False, None) if \(j\) is not a CM j-invariant or (True, (d,f)) if \(j\) is the \(j\)-invariant of the imaginary quadratic order of discriminant \(D=df^2\) where \(d\) is the associated fundamental discriminant and \(f\) the index.

Note

The current implementation makes use of the classification of all orders of class number up to 100, and hence will raise an error if \(j\) is an algebraic integer of degree greater than this. It would be possible to implement a more general version, using the fact that \(d\) must be supported on the primes dividing the discriminant of the minimal polynomial of \(j\).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: is_cm_j_invariant(0)
(True, (-3, 1))
sage: is_cm_j_invariant(8000)
(True, (-8, 1))

sage: K.<a> = QuadraticField(5)
sage: is_cm_j_invariant(282880*a + 632000)
(True, (-20, 1))
sage: K.<a> = NumberField(x^3 - 2)
sage: is_cm_j_invariant(31710790944000*a^2 + 39953093016000*a + 50337742902000)
(True, (-3, 6))

sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(h)#

Return largest absolute value of any fundamental discriminant with class number \(h\), and the number of fundamental discriminants with that class number. This is known for \(h\) up to 100, by work of Mark Watkins.

INPUT:

\(h\) – integer

EXAMPLES:

sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(0)
(0, 0)
sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(1)
(163, 9)
sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(2)
(427, 18)
sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(10)
(13843, 87)
sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(100)
(1856563, 1736)
sage: sage.schemes.elliptic_curves.cm.largest_fundamental_disc_with_class_number(101)
Traceback (most recent call last):
...
NotImplementedError: largest discriminant not known for class number 101