Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 5, -3, -2, 1]))
gp: K = nfinit(Polrev([-1, 1, 5, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,7,-7,-3,2]),K([2,6,-3,-2,1]),K([-2,0,1,0,0]),K([-76,-100,115,46,-32]),K([97,132,-144,-60,40])])
gp: E = ellinit([Polrev([4,7,-7,-3,2]),Polrev([2,6,-3,-2,1]),Polrev([-2,0,1,0,0]),Polrev([-76,-100,115,46,-32]),Polrev([97,132,-144,-60,40])], K);
magma: E := EllipticCurve([K![4,7,-7,-3,2],K![2,6,-3,-2,1],K![-2,0,1,0,0],K![-76,-100,115,46,-32],K![97,132,-144,-60,40]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-4a^2+6a+2)\) | = | \((a^4-2a^3-4a^2+6a+2)\) |
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: | \((-2a^4+6a^3+3a^2-13a)\) | = | \((a^4-2a^3-4a^2+6a+2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2401 \) | = | \(49^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4128570706}{49} a^{4} - \frac{2238511927}{49} a^{3} - \frac{4911331220}{49} a^{2} + \frac{180417913}{49} a + \frac{791318512}{49} \) | ||
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 | $\left(6 a^{4} - 9 a^{3} - 22 a^{2} + 19 a + 15 : 13 a^{4} - 21 a^{3} - 48 a^{2} + 46 a + 33 : 1\right)$ | |
| Height | \(0.032942691141070157579231785036761819669\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3 : -4 a^{4} + 6 a^{3} + 14 a^{2} - 13 a - 9 : 1\right)$ | $\left(\frac{1}{2} a^{4} - \frac{1}{4} a^{3} - 2 a^{2} - \frac{1}{4} a + 2 : -\frac{13}{8} a^{4} + \frac{19}{8} a^{3} + \frac{41}{8} a^{2} - \frac{11}{2} a - \frac{17}{8} : 1\right)$ |
| 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: | \( 0.032942691141070157579231785036761819669 \) | ||
| Period: | \( 19054.756907846291959484785938570314353 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.05359085 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-4a^2+6a+2)\) | \(49\) | \(2\) | \(I_{2}\) | Non-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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
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.