Base field \(\Q(\sqrt{13}) \)
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([1,0]),K([0,-1]),K([1,1]),K([-11530,-1005]),K([-27456,288310])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([1,1]),Polrev([-11530,-1005]),Polrev([-27456,288310])], K);
magma: E := EllipticCurve([K![1,0],K![0,-1],K![1,1],K![-11530,-1005],K![-27456,288310]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-13a+52)\) | = | \((-a+1)^{2}\cdot(-2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1521 \) | = | \(3^{2}\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-221a+299)\) | = | \((-a+1)^{6}\cdot(-2a+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 123201 \) | = | \(3^{6}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 3387888351672962316333 a - 7801546759588524979828 \) | ||
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 | $\left(-\frac{7514}{25} a + \frac{18756}{25} : -\frac{1319413}{125} a + \frac{3016219}{125} : 1\right)$ |
| Height | \(7.9425179706494976755447990783746618123\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 7.9425179706494976755447990783746618123 \) | ||
| Period: | \( 0.28191694325206129971073457517173248139 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.4840921312061201779277285373531970041 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((-2a+1)\) | \(13\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(19\) | 19B.8.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 19 and 57.
Its isogeny class
1521.3-e
consists of curves linked by isogenies of
degrees dividing 57.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.