Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp: K = nfinit(Pol(Vecrev([2, -1, 1])));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([6176,0]),K([-69388,0])])
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([6176,0])),Pol(Vecrev([-69388,0]))], K);
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![6176,0],K![-69388,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-96 a + 48)\) | = | \( \left(a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(3\right) \cdot \left(-2 a + 1\right) \) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16128 \) | = | \( 2^{8} \cdot 7 \cdot 9 \) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((17279298183168)\) | = | \( \left(a\right)^{13} \cdot \left(-a + 1\right)^{13} \cdot \left(3\right)^{16} \cdot \left(-2 a + 1\right)^{4} \) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 298574145702832945678516224 \) | = | \( 2^{26} \cdot 7^{4} \cdot 9^{16} \) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{6359387729183}{4218578658} \) | ||
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 | $\left(-\frac{10935}{121} a - \frac{127}{121} : -\frac{87480}{1331} a - \frac{1599426}{1331} : 1\right)$ |
| Height | \(2.31081181470693\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(92 : -1134 : 1\right)$ |
| 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: | \( 2.31081181470693 \) | ||
| Period: | \( 0.0856364791332518 \) | ||
| Tamagawa product: | \( 256 \) = \(2\cdot2\cdot2^{2}\cdot2^{4}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \(4.78689979783825\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a\right) \) | \(2\) | \(2\) | \(I_{5}^*\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
| \( \left(-a + 1\right) \) | \(2\) | \(2\) | \(I_{5}^*\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
| \( \left(-2 a + 1\right) \) | \(7\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \( \left(3\right) \) | \(9\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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, 4, 8 and 16.
Its isogeny class
16128.5-n
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This curve is the base change of elliptic curves 2352.l6, 336.d6, defined over \(\Q\), so it is also a \(\Q\)-curve.