Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([1,0]),K([-1579,206]),K([-24011,3981])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([1,0]),Polrev([-1579,206]),Polrev([-24011,3981])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![1,0],K![-1579,206],K![-24011,3981]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-15a+60)\) | = | \((-a)\cdot(a-1)^{2}\cdot(-a-1)^{2}\cdot(a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 3375 \) | = | \(3\cdot3^{2}\cdot5^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-692435625a+29608125)\) | = | \((-a)\cdot(a-1)^{18}\cdot(-a-1)^{9}\cdot(a-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1418776204833984375 \) | = | \(3\cdot3^{18}\cdot5^{9}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{59052841710247}{332150625} a + \frac{4469076589604}{110716875} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(\frac{869}{81} a - \frac{4589}{81} : \frac{132554}{729} a + \frac{92866}{729} : 1\right)$ |
| Height | \(3.1045184897929029496757996297733690238\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{3}{4} a - \frac{89}{4} : \frac{43}{4} a + \frac{5}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 3.1045184897929029496757996297733690238 \) | ||
| Period: | \( 0.29565495520933112193952882115942461557 \) | ||
| Tamagawa product: | \( 16 \) = \(1\cdot2^{2}\cdot2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 4.4279535157085050612334338526147266451 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a-1)\) | \(3\) | \(4\) | \(I_{12}^{*}\) | Additive | \(-1\) | \(2\) | \(18\) | \(12\) |
| \((-a-1)\) | \(5\) | \(2\) | \(I_{3}^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(3\) |
| \((a-2)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
3375.10-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.