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([0,0]),K([-1,0]),K([0,0]),K([-17,4]),K([-171,40])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([-17,4]),Polrev([-171,40])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![-17,4],K![-171,40]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((16)\) | = | \((a-4)^{4}\cdot(a+3)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 256 \) | = | \(2^{4}\cdot2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-256a-1280)\) | = | \((a-4)^{8}\cdot(a+3)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1048576 \) | = | \(2^{8}\cdot2^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -7168 a - 23552 \) | ||
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 | $\left(-\frac{138048}{29929} a + \frac{679988}{29929} : -\frac{152317382}{5177717} a + \frac{673315513}{5177717} : 1\right)$ |
| Height | \(7.5645544594144469606325989392136522237\) |
| 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: | \( 7.5645544594144469606325989392136522237 \) | ||
| Period: | \( 2.9887855612822844759022866761489520932 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 5.9892256816147320882963115863162467836 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-4)\) | \(2\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(4\) | \(8\) | \(0\) |
| \((a+3)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(4\) | \(12\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 256.1-n consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.