Base field 5.5.38569.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,5,0,-1]),K([4,-5,-5,1,1]),K([1,0,0,0,0]),K([-453,165,608,-33,-123]),K([-3989,1501,5403,-306,-1097])])
gp: E = ellinit([Polrev([-3,0,5,0,-1]),Polrev([4,-5,-5,1,1]),Polrev([1,0,0,0,0]),Polrev([-453,165,608,-33,-123]),Polrev([-3989,1501,5403,-306,-1097])], K);
magma: E := EllipticCurve([K![-3,0,5,0,-1],K![4,-5,-5,1,1],K![1,0,0,0,0],K![-453,165,608,-33,-123],K![-3989,1501,5403,-306,-1097]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^4-a^3+9a^2+3a-5)\) | = | \((-2a^4-a^3+9a^2+3a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 73 \) | = | \(73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-6a^4+a^3+25a^2+6a+2)\) | = | \((-2a^4-a^3+9a^2+3a-5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 389017 \) | = | \(73^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4005895010964759}{389017} a^{4} + \frac{8209363168524430}{389017} a^{3} + \frac{3566257207321731}{389017} a^{2} - \frac{8377320503431615}{389017} a + \frac{1890337860286333}{389017} \) | ||
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: | 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 7.9112222232547539407054853177439350154 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.08764748 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^4-a^3+9a^2+3a-5)\) | \(73\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
73.1-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.