Base field 4.4.17428.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + 4 x + 6 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([6, 4, -6, -1, 1]))
gp: K = nfinit(Polrev([6, 4, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, 4, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-4,1,1]),K([1,-5,-1,1]),K([0,-4,0,1]),K([-357,-346,180,125]),K([-19415,-20942,10450,7601])])
gp: E = ellinit([Polrev([-3,-4,1,1]),Polrev([1,-5,-1,1]),Polrev([0,-4,0,1]),Polrev([-357,-346,180,125]),Polrev([-19415,-20942,10450,7601])], K);
magma: E := EllipticCurve([K![-3,-4,1,1],K![1,-5,-1,1],K![0,-4,0,1],K![-357,-346,180,125],K![-19415,-20942,10450,7601]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-a-2)\) | = | \((a^3+a^2-3a-2)\cdot(-a^2+2a+1)\) |
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: | \((32768a^2-16384a-81920)\) | = | \((a^3+a^2-3a-2)^{42}\cdot(-a^2+2a+1)^{21}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -9223372036854775808 \) | = | \(-2^{42}\cdot2^{21}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{10495049249}{2097152} a^{3} - \frac{520166291}{524288} a^{2} + \frac{27328674037}{1048576} a + \frac{10950681647}{1048576} \) | ||
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: | $\left(-\frac{9}{4} a^{3} - 5 a^{2} + \frac{9}{2} a + \frac{23}{2} : \frac{75}{8} a^{3} + \frac{27}{2} a^{2} - \frac{103}{4} a - \frac{105}{4} : 1\right)$ |
| 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: | \( 8.6405914662029696245130537416369278238 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.294531812505927 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-3a-2)\) | \(2\) | \(2\) | \(I_{42}\) | Non-split multiplicative | \(1\) | \(1\) | \(42\) | \(42\) |
| \((-a^2+2a+1)\) | \(2\) | \(1\) | \(I_{21}\) | Non-split multiplicative | \(1\) | \(1\) | \(21\) | \(21\) |
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 and 6.
Its isogeny class
4.2-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.