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([0,0]),K([0,0]),K([-8,0]),K([0,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-8,0]),Polrev([0,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-8,0],K![0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-4a+2)\) | = | \((a-4)\cdot(a+3)\cdot(4a+13)\cdot(10a-43)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 228 \) | = | \(2\cdot2\cdot3\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((32832)\) | = | \((a-4)^{6}\cdot(a+3)^{6}\cdot(4a+13)^{6}\cdot(10a-43)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1077940224 \) | = | \(2^{6}\cdot2^{6}\cdot3^{6}\cdot19^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{57066625}{32832} \) | ||
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: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/6\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(0 : 0 : 1\right)$ | $\left(-2 a + 8 : -6 a + 28 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 9.9153867679137740798773190912464473945 \) | ||
| Tamagawa product: | \( 432 \) = \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\) | ||
| Torsion order: | \(12\) | ||
| Leading coefficient: | \( 3.9399751820749132960245004898835962503 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-4)\) | \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((a+3)\) | \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((4a+13)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((10a-43)\) | \(19\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
228.1-h
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 114.c3 |
| \(\Q\) | 6498.t3 |