Base field \(\Q(\sqrt{11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))
gp: K = nfinit(Polrev([-11, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([-14864,4488]),K([-977736,294804])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-14864,4488]),Polrev([-977736,294804])], K);
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![-14864,4488],K![-977736,294804]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((14a+42)\) | = | \((a+3)^{3}\cdot(a+2)\cdot(a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 392 \) | = | \(2^{3}\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((224)\) | = | \((a+3)^{10}\cdot(a+2)\cdot(a-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 50176 \) | = | \(2^{10}\cdot7\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1443468546}{7} \) | ||
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 | $\left(-\frac{3947}{25} a + \frac{13018}{25} : -\frac{648772}{125} a + \frac{2152518}{125} : 1\right)$ |
| Height | \(3.2870054824984663221636800781952485071\) |
| 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: | $\left(\frac{29}{2} a - 51 : \frac{73}{4} a - \frac{217}{4} : 1\right)$ |
| 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: | \( 3.2870054824984663221636800781952485071 \) | ||
| Period: | \( 2.6362935287269056242494966545455530267 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 5.2254999164207512700223916768533243095 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((a+2)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a-2)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
392.1-f
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 112.b1 |
| \(\Q\) | 6776.g1 |