Base field 4.4.15317.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([2, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-4,-1,1]),K([5,-2,-2,1]),K([-1,-5,0,1]),K([188,341,-549,138]),K([3045,5861,-9401,2457])])
gp: E = ellinit([Polrev([1,-4,-1,1]),Polrev([5,-2,-2,1]),Polrev([-1,-5,0,1]),Polrev([188,341,-549,138]),Polrev([3045,5861,-9401,2457])], K);
magma: E := EllipticCurve([K![1,-4,-1,1],K![5,-2,-2,1],K![-1,-5,0,1],K![188,341,-549,138],K![3045,5861,-9401,2457]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2-a-4)\) | = | \((-a)\cdot(a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 4 \) | = | \(2\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((292a^3-973a^2-627a+2762)\) | = | \((-a)^{2}\cdot(a-1)^{36}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 274877906944 \) | = | \(2^{2}\cdot2^{36}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{25244062543252502649437931}{68719476736} a^{3} - \frac{58811778546102253618393565}{68719476736} a^{2} - \frac{81584408494612303988948299}{68719476736} a + \frac{76560454679459868202267199}{34359738368} \) | ||
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 | $\left(2 a^{3} - \frac{31}{4} a^{2} + \frac{3}{4} a + \frac{21}{4} : \frac{25}{4} a^{3} - \frac{135}{8} a^{2} - \frac{1}{8} a + \frac{99}{8} : 1\right)$ | |
Height | \(2.5691624343281407373827256376538691091\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
Torsion generators: | $\left(\frac{5}{4} a^{3} - \frac{27}{4} a^{2} + \frac{23}{4} a - \frac{1}{2} : \frac{9}{8} a^{3} - \frac{7}{8} a^{2} - \frac{25}{8} a + \frac{9}{4} : 1\right)$ | $\left(a^{3} - 5 a^{2} + 2 a + 1 : a^{3} - 3 a - 1 : 1\right)$ |
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: | \( 2.5691624343281407373827256376538691091 \) | ||
Period: | \( 13.055766083325680657798104451444772571 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 2.43921216239754 \) | ||
Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a-1)\) | \(2\) | \(2\) | \(I_{36}\) | Non-split multiplicative | \(1\) | \(1\) | \(36\) | \(36\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
\(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
4.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.