Base field \(\Q(\sqrt{57}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))
gp: K = nfinit(Polrev([-14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([-27,-9]),K([-81,-25])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([-27,-9]),Polrev([-81,-25])], K);
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![-27,-9],K![-81,-25]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((14)\) | = | \((a-4)\cdot(a+3)\cdot(2a+7)\cdot(-2a+9)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 196 \) | = | \(2\cdot2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((114464a-410816)\) | = | \((a-4)^{14}\cdot(a+3)^{5}\cdot(2a+7)^{2}\cdot(-2a+9)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -61681958912 \) | = | \(-2^{14}\cdot2^{5}\cdot7^{2}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{9459305507}{39337984} a + \frac{4425485353}{5619712} \) | ||
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 | $\left(-a + 1 : 6 : 1\right)$ |
| Height | \(0.026453311637233377099900253237847708259\) |
| 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: | $\left(-\frac{1}{4} a - \frac{3}{2} : -\frac{3}{8} a + \frac{1}{4} : 1\right)$ |
| 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: | \( 0.026453311637233377099900253237847708259 \) | ||
| Period: | \( 6.3393819569556048331031849512069994056 \) | ||
| Tamagawa product: | \( 560 \) = \(( 2 \cdot 7 )\cdot5\cdot2\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 6.2193868516085390385091948436835335378 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-4)\) | \(2\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
| \((a+3)\) | \(2\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((2a+7)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-2a+9)\) | \(7\) | \(4\) | \(I_{4}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
196.1-g
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.