Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0,0]),K([2,4,-9,-1,2]),K([0,1,-4,0,1]),K([-5,-21,-25,5,5]),K([280,419,-1003,-145,222])])
gp: E = ellinit([Polrev([1,1,0,0,0]),Polrev([2,4,-9,-1,2]),Polrev([0,1,-4,0,1]),Polrev([-5,-21,-25,5,5]),Polrev([280,419,-1003,-145,222])], K);
magma: E := EllipticCurve([K![1,1,0,0,0],K![2,4,-9,-1,2],K![0,1,-4,0,1],K![-5,-21,-25,5,5],K![280,419,-1003,-145,222]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-a^3-5a^2+5a+4)\) | = | \((a^4-a^3-5a^2+5a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 97 \) | = | \(97\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-14a^4+17a^3+74a^2-46a-82)\) | = | \((a^4-a^3-5a^2+5a+4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -88529281 \) | = | \(-97^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{298244004312035170314}{88529281} a^{4} - \frac{224778495680070789765}{88529281} a^{3} - \frac{1183654637007985475374}{88529281} a^{2} + \frac{271982011051561202569}{88529281} a + \frac{648707452832843494777}{88529281} \) | ||
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: | $\left(-6 a^{4} + 3 a^{3} + \frac{107}{4} a^{2} - \frac{13}{2} a - \frac{25}{4} : a^{4} + \frac{1}{8} a^{3} - \frac{41}{8} a^{2} - \frac{25}{8} a + \frac{1}{8} : 1\right)$ |
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: | \( 9.3610492402523353763895353185589322852 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 0.962463662 \) | ||
Analytic order of Ш: | \( 16 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-a^3-5a^2+5a+4)\) | \(97\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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
97.1-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.