Base field 5.5.81509.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 5 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 3, -5, -1, 1]))
gp: K = nfinit(Polrev([-2, 5, 3, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 3, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-1,1,0]),K([-4,1,5,0,-1]),K([3,2,-4,-1,1]),K([2,-2,1,2,-1]),K([-5,3,4,-2,0])])
gp: E = ellinit([Polrev([2,-3,-1,1,0]),Polrev([-4,1,5,0,-1]),Polrev([3,2,-4,-1,1]),Polrev([2,-2,1,2,-1]),Polrev([-5,3,4,-2,0])], K);
magma: E := EllipticCurve([K![2,-3,-1,1,0],K![-4,1,5,0,-1],K![3,2,-4,-1,1],K![2,-2,1,2,-1],K![-5,3,4,-2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-a-1)\) | = | \((a^2-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^2+a+1)\) | = | \((a^2-a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -9 \) | = | \(-9\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -53060 a^{4} + \frac{432928}{3} a^{3} + 15463 a^{2} - \frac{565285}{3} a + \frac{188426}{3} \) | ||
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 | $\left(a^{3} - a^{2} - 4 a + 3 : -a^{4} + 2 a^{3} + 3 a^{2} - 6 a : 1\right)$ |
| Height | \(0.0053535046239124893469020718371728636340\) |
| 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: | \( 0.0053535046239124893469020718371728636340 \) | ||
| Period: | \( 18888.185724521271011073835923308402105 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.77090657 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-a-1)\) | \(9\) | \(1\) | \(I_{1}\) | Non-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 |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
9.1-a
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.