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([0,3,1,-1]),K([2,-4,-1,1]),K([-867,615,305,-213]),K([-12175,8409,4341,-2875])])
gp: E = ellinit([Polrev([1,-4,-1,1]),Polrev([0,3,1,-1]),Polrev([2,-4,-1,1]),Polrev([-867,615,305,-213]),Polrev([-12175,8409,4341,-2875])], K);
magma: E := EllipticCurve([K![1,-4,-1,1],K![0,3,1,-1],K![2,-4,-1,1],K![-867,615,305,-213],K![-12175,8409,4341,-2875]]);
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: | \((-13a^3+26a^2+41a-54)\) | = | \((-a)^{8}\cdot(a-1)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 131072 \) | = | \(2^{8}\cdot2^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{280401766250925363}{512} a^{3} + \frac{1034629287445097955}{512} a^{2} - \frac{629865483106281211}{512} a - \frac{20638781304796925}{32} \) | ||
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 | $\left(\frac{91}{2} a^{3} - \frac{367}{4} a^{2} - 132 a + 259 : \frac{4467}{8} a^{3} - \frac{6433}{4} a^{2} - \frac{7373}{4} a + \frac{8405}{2} : 1\right)$ |
| Height | \(2.5691624343281407373827256376538691091\) |
| 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: | $\left(\frac{17}{2} a^{3} - 12 a^{2} - \frac{103}{4} a + \frac{61}{2} : -\frac{63}{8} a^{3} + \frac{23}{2} a^{2} + \frac{41}{2} a - 37 : 1\right)$ |
| 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: | \( 1.6319707604157100822247630564305965714 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{3}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((a-1)\) | \(2\) | \(1\) | \(I_{9}\) | Non-split multiplicative | \(1\) | \(1\) | \(9\) | \(9\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 8, 12 and 24.
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.