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([1,6,-3,-2,1]),K([-4,-5,4,2,-1]),K([2,3,-4,-1,1]),K([-20,55,-11,-14,-5]),K([-100,193,166,-115,-63])])
gp: E = ellinit([Polrev([1,6,-3,-2,1]),Polrev([-4,-5,4,2,-1]),Polrev([2,3,-4,-1,1]),Polrev([-20,55,-11,-14,-5]),Polrev([-100,193,166,-115,-63])], K);
magma: E := EllipticCurve([K![1,6,-3,-2,1],K![-4,-5,4,2,-1],K![2,3,-4,-1,1],K![-20,55,-11,-14,-5],K![-100,193,166,-115,-63]]);
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: | \((-17a^4+78a^3+26a^2-333a+5)\) | = | \((a^4-2a^3-2a^2+5a)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -152587890625 \) | = | \(-25^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{136402392772743238}{390625} a^{4} + \frac{341339187700279242}{390625} a^{3} + \frac{236740655090238193}{390625} a^{2} - \frac{799831738588792786}{390625} a + \frac{268640638721913956}{390625} \) | ||
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: | \(0\) |
| 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: | $\left(\frac{3}{4} a^{4} - \frac{7}{4} a^{3} - 5 a^{2} + \frac{13}{2} a : -\frac{3}{8} a^{4} + \frac{3}{2} a^{3} + \frac{3}{2} a^{2} + \frac{11}{8} a - \frac{37}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 27.648888785496221433508058684067197501 \) | ||
| Tamagawa product: | \( 8 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.15781477 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-2a^2+5a)\) | \(25\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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
25.1-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.