Base field 4.4.13025.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 3 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 3, -12, -1, 1]))
gp: K = nfinit(Polrev([29, 3, -12, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 3, -12, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([9/4,-3/2,-1/2,1/4]),K([-1,-1,0,0]),K([-5,0,1,0]),K([-5/2,12,1,-7/2]),K([-177/4,-65/2,27/2,39/4])])
gp: E = ellinit([Polrev([9/4,-3/2,-1/2,1/4]),Polrev([-1,-1,0,0]),Polrev([-5,0,1,0]),Polrev([-5/2,12,1,-7/2]),Polrev([-177/4,-65/2,27/2,39/4])], K);
magma: E := EllipticCurve([K![9/4,-3/2,-1/2,1/4],K![-1,-1,0,0],K![-5,0,1,0],K![-5/2,12,1,-7/2],K![-177/4,-65/2,27/2,39/4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/4a^3+1/2a^2-1/2a-7/4)\) | = | \((1/4a^3+1/2a^2-1/2a-7/4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3/4a^3+3/2a^2+11/2a-19/4)\) | = | \((1/4a^3+1/2a^2-1/2a-7/4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{5243081}{20} a^{3} - \frac{2114437}{10} a^{2} + \frac{27643383}{10} a + \frac{84189419}{20} \) | ||
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 | $\left(-\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a + \frac{3}{4} : -\frac{3}{4} a^{3} - \frac{1}{2} a^{2} + \frac{11}{2} a + \frac{33}{4} : 1\right)$ |
| Height | \(0.27900112998539265966839896484009021171\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{1}{4} a^{3} + \frac{5}{2} a^{2} + \frac{3}{2} a - \frac{33}{4} : \frac{1}{4} a^{3} - \frac{17}{2} a^{2} - \frac{3}{2} a + \frac{121}{4} : 1\right)$ |
| 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: | \( 0.27900112998539265966839896484009021171 \) | ||
| Period: | \( 1484.2514728195000566936756381707098341 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.81423937825073 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/4a^3+1/2a^2-1/2a-7/4)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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, 4 and 8.
Its isogeny class
5.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.