Base field \(\Q(\sqrt{53}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 13 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-13, -1, 1]))
gp: K = nfinit(Polrev([-13, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([1,1]),K([-354439,-112430]),K([-122894996,-39130752])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([1,1]),Polrev([-354439,-112430]),Polrev([-122894996,-39130752])], K);
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,1],K![-354439,-112430],K![-122894996,-39130752]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-11a-15)\) | = | \((-a-2)\cdot(a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1183 \) | = | \(7\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1597a+726)\) | = | \((-a-2)\cdot(a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -33787663 \) | = | \(-7\cdot13^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{195338235078135808}{7} a - \frac{613353587397607424}{7} \) | ||
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 | $\left(\frac{53384}{363} a + \frac{477788}{1089} : -\frac{7672201}{35937} a - \frac{24052609}{35937} : 1\right)$ |
| Height | \(7.4746374867256631890062550200598293566\) |
| 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: | \( 7.4746374867256631890062550200598293566 \) | ||
| Period: | \( 0.061823157247357731537915546162261688138 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 6.2205596044844117669849126804995850209 \) | ||
| Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-2)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a-1)\) | \(13\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
1183.3-g
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.