Base field 4.4.19796.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, 1, -7, -1, 1]))
gp: K = nfinit(Polrev([8, 1, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 1, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([3,-4,-2,1]),K([0,1,0,0]),K([257,110,-193,-97]),K([6149,2845,-4424,-2269])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([3,-4,-2,1]),Polrev([0,1,0,0]),Polrev([257,110,-193,-97]),Polrev([6149,2845,-4424,-2269])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![3,-4,-2,1],K![0,1,0,0],K![257,110,-193,-97],K![6149,2845,-4424,-2269]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a+2)\) | = | \((-a^2+2)\cdot(a^2+a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a^3-31a^2+19a+96)\) | = | \((-a^2+2)^{3}\cdot(a^2+a-1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -625000 \) | = | \(-2^{3}\cdot5^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1267618337281}{625000} a^{3} + \frac{488374808849}{625000} a^{2} + \frac{2101610388979}{125000} a + \frac{9997063509159}{625000} \) | ||
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 | $\left(-2 a^{3} - a^{2} + 7 a + 5 : 2 a^{3} + a^{2} - 11 a - 11 : 1\right)$ |
| Height | \(0.52303180715897640577380623902916810853\) |
| 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: | $\left(-\frac{5}{4} a^{3} - \frac{3}{4} a^{2} + \frac{7}{4} a - \frac{5}{4} : \frac{5}{8} a^{3} + \frac{3}{8} a^{2} - \frac{11}{8} a + \frac{5}{8} : 1\right)$ |
| 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: | \( 0.52303180715897640577380623902916810853 \) | ||
| Period: | \( 118.20842322784249470577998319277210348 \) | ||
| Tamagawa product: | \( 7 \) = \(1\cdot7\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.07599653732709 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+2)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((a^2+a-1)\) | \(5\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
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
10.2-d
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.