Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,3,-4,-1,1]),K([-2,0,5,0,-1]),K([-3,-2,5,1,-1]),K([27,92,-65,-34,18]),K([-65,-235,183,87,-50])])
gp: E = ellinit([Polrev([3,3,-4,-1,1]),Polrev([-2,0,5,0,-1]),Polrev([-3,-2,5,1,-1]),Polrev([27,92,-65,-34,18]),Polrev([-65,-235,183,87,-50])], K);
magma: E := EllipticCurve([K![3,3,-4,-1,1],K![-2,0,5,0,-1],K![-3,-2,5,1,-1],K![27,92,-65,-34,18],K![-65,-235,183,87,-50]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+3a^2-2a)\) | = | \((-a^2+a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^4+8a^3+30a^2-13a-16)\) | = | \((-a^2+a+2)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -78125 \) | = | \(-5^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \) | ||
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 | $\left(a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 2 : -4 a^{4} + 9 a^{3} + 14 a^{2} - 26 a - 6 : 1\right)$ |
| Height | \(0.0022474568430428602391936235774522493923\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0022474568430428602391936235774522493923 \) | ||
| Period: | \( 8560.3931330344619768579587207605095591 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.50167015 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+a+2)\) | \(5\) | \(4\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 25.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.