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([-1,0,1,0,0]),K([0,3,1,-1,0]),K([-2,-1,1,0,0]),K([19,4,-95,-5,19]),K([-17,30,182,38,-52])])
gp: E = ellinit([Polrev([-1,0,1,0,0]),Polrev([0,3,1,-1,0]),Polrev([-2,-1,1,0,0]),Polrev([19,4,-95,-5,19]),Polrev([-17,30,182,38,-52])], K);
magma: E := EllipticCurve([K![-1,0,1,0,0],K![0,3,1,-1,0],K![-2,-1,1,0,0],K![19,4,-95,-5,19],K![-17,30,182,38,-52]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+4a^3-10a+1)\) | = | \((a^2-1)\cdot(a^3-3a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 507 \) | = | \(3\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-65a^4+60a^3+177a^2+17a+85)\) | = | \((a^2-1)^{8}\cdot(a^3-3a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -31668693849 \) | = | \(-3^{8}\cdot13^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{7766624836025326355}{6561} a^{4} + \frac{6488464292861171239}{2187} a^{3} + \frac{4481598027790652077}{2187} a^{2} - \frac{45640052133196144267}{6561} a + \frac{15340352105491554221}{6561} \) | ||
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 | $\left(a^{4} - 3 a^{3} - a^{2} + 5 a + 4 : 7 a^{4} - 12 a^{3} - 22 a^{2} + 22 a + 16 : 1\right)$ |
| Height | \(0.073567810663153802471279723413290363196\) |
| 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: | $\left(-\frac{9}{4} a^{4} + 3 a^{3} + \frac{15}{2} a^{2} - 5 a - \frac{9}{4} : \frac{5}{8} a^{3} - \frac{1}{2} a^{2} - \frac{13}{8} a + \frac{5}{8} : 1\right)$ |
| 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: | \( 0.073567810663153802471279723413290363196 \) | ||
| Period: | \( 1725.9630798290260225889278814933342368 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.32323271 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-1)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((a^3-3a)\) | \(13\) | \(2\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
507.1-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.