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([3,7,-7,-3,2]),K([1,0,-1,0,0]),K([-1,5,-2,-2,1]),K([-7,40,15,-21,-8]),K([-9,-1,111,-10,-51])])
gp: E = ellinit([Polrev([3,7,-7,-3,2]),Polrev([1,0,-1,0,0]),Polrev([-1,5,-2,-2,1]),Polrev([-7,40,15,-21,-8]),Polrev([-9,-1,111,-10,-51])], K);
magma: E := EllipticCurve([K![3,7,-7,-3,2],K![1,0,-1,0,0],K![-1,5,-2,-2,1],K![-7,40,15,-21,-8],K![-9,-1,111,-10,-51]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-2a^2+5a)\) | = | \((a^4-2a^3-2a^2+5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((10a^4-18a^3-37a^2+42a+16)\) | = | \((a^4-2a^3-2a^2+5a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 390625 \) | = | \(25^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1549868991089}{625} a^{4} - \frac{2463604541892}{625} a^{3} - \frac{5660845609864}{625} a^{2} + \frac{5425940335044}{625} a + \frac{3777390596864}{625} \) | ||
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 | $\left(\frac{25}{9} a^{4} + \frac{13}{3} a^{3} - \frac{11}{3} a^{2} - \frac{58}{9} a + \frac{5}{9} : -\frac{1336}{27} a^{4} - \frac{172}{9} a^{3} + \frac{971}{9} a^{2} + \frac{406}{27} a - \frac{449}{27} : 1\right)$ | |
| Height | \(0.40039808757450243528609034976141255487\) | |
| 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: | $\left(2 a^{3} + a^{2} - 4 a : -a^{4} + a^{2} - a + 3 : 1\right)$ | $\left(-2 a^{3} + a^{2} + 4 a - 4 : 3 a^{4} - 4 a^{3} - 9 a^{2} + 9 a + 3 : 1\right)$ |
| 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: | \( 0.40039808757450243528609034976141255487 \) | ||
| Period: | \( 1346.7964560078881861072817568643666563 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.76419015 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-2a^2+5a)\) | \(25\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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
25.1-d
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.