Base field 4.4.17725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 13, -12, -2, 1]))
gp: K = nfinit(Polrev([41, 13, -12, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-7,0,1,0]),K([1,-8,-1,1]),K([-6,-8,0,1]),K([-41,5,8,-1]),K([-31,-25,2,3])])
gp: E = ellinit([Polrev([-7,0,1,0]),Polrev([1,-8,-1,1]),Polrev([-6,-8,0,1]),Polrev([-41,5,8,-1]),Polrev([-31,-25,2,3])], K);
magma: E := EllipticCurve([K![-7,0,1,0],K![1,-8,-1,1],K![-6,-8,0,1],K![-41,5,8,-1],K![-31,-25,2,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-10)\) | = | \((a^2-10)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^3-9a^2-9a+13)\) | = | \((a^2-10)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 117649 \) | = | \(49^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{72741807}{343} a^{3} + 1132866 a^{2} + \frac{41876676}{343} a - \frac{1756222938}{343} \) | ||
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 | $\left(-a^{3} + 4 a^{2} + 5 a - 22 : -3 a^{3} + 4 a^{2} + 19 a - 15 : 1\right)$ |
| Height | \(0.26948882381287684799660457606920475597\) |
| 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: | $\left(a^{2} - 6 : a^{3} - a^{2} - 6 a + 3 : 1\right)$ |
| 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: | \( 0.26948882381287684799660457606920475597 \) | ||
| Period: | \( 1254.7822675607785907684262266455203444 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.90492233135250 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-10)\) | \(49\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 and 4.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.