Base field 5.5.181057.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} + 2 x - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 2, 7, -4, -2, 1]))
gp: K = nfinit(Polrev([-3, 2, 7, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 2, 7, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-1,-5,0,1]),K([-2,2,2,-1,0]),K([-1,-1,1,0,0]),K([-24,-7,30,3,-5]),K([7,-1,4,1,-1])])
gp: E = ellinit([Polrev([3,-1,-5,0,1]),Polrev([-2,2,2,-1,0]),Polrev([-1,-1,1,0,0]),Polrev([-24,-7,30,3,-5]),Polrev([7,-1,4,1,-1])], K);
magma: E := EllipticCurve([K![3,-1,-5,0,1],K![-2,2,2,-1,0],K![-1,-1,1,0,0],K![-24,-7,30,3,-5],K![7,-1,4,1,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-2a^2-3a+4)\) | = | \((a^3-2a^2-3a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^2+a+4)\) | = | \((a^3-2a^2-3a+4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -25 \) | = | \(-25\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{48803087}{5} a^{4} + \frac{185083624}{5} a^{3} - 27082805 a^{2} - \frac{101317246}{5} a + \frac{82622024}{5} \) | ||
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 | $\left(-2 : 2 a^{4} - a^{3} - 11 a^{2} + 2 a + 10 : 1\right)$ |
| Height | \(0.014349913243652449996884116134319860133\) |
| 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: | \( 0.014349913243652449996884116134319860133 \) | ||
| Period: | \( 21629.269713221235786299715393757905384 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.64714875 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-2a^2-3a+4)\) | \(25\) | \(1\) | \(I_{1}\) | 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
25.1-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.