Base field 4.4.4205.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,3,2,-1]),K([1,-7,-3,2]),K([-1,4,2,-1]),K([-3,9,3,-2]),K([4,-12,-5,3])])
gp: E = ellinit([Polrev([0,3,2,-1]),Polrev([1,-7,-3,2]),Polrev([-1,4,2,-1]),Polrev([-3,9,3,-2]),Polrev([4,-12,-5,3])], K);
magma: E := EllipticCurve([K![0,3,2,-1],K![1,-7,-3,2],K![-1,4,2,-1],K![-3,9,3,-2],K![4,-12,-5,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a^2-a+3)\) | = | \((a^3-a^2-5a)\cdot(a^2-2a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 35 \) | = | \(5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-8a-3)\) | = | \((a^3-a^2-5a)^{2}\cdot(a^2-2a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -175 \) | = | \(-5^{2}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{324614}{35} a^{3} + \frac{777407}{35} a^{2} + \frac{371417}{35} a - \frac{212344}{35} \) | ||
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 | $\left(-a^{3} + 6 a + 5 : -a^{3} - a^{2} + 7 a + 10 : 1\right)$ |
| Height | \(0.21337390952776005525302766484456271456\) |
| 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: | $\left(-2 a^{3} + 3 a^{2} + 8 a - 2 : -1 : 1\right)$ |
| 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: | \( 0.21337390952776005525302766484456271456 \) | ||
| Period: | \( 319.13547734805841205893553406633278640 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.10021326092434 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-5a)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((a^2-2a-3)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.
Its isogeny class
35.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.