Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-3,0,1]),K([0,-4,-1,1]),K([0,0,0,0]),K([161,101,-192,49]),K([-919,217,1819,-755])])
gp: E = ellinit([Polrev([0,-3,0,1]),Polrev([0,-4,-1,1]),Polrev([0,0,0,0]),Polrev([161,101,-192,49]),Polrev([-919,217,1819,-755])], K);
magma: E := EllipticCurve([K![0,-3,0,1],K![0,-4,-1,1],K![0,0,0,0],K![161,101,-192,49],K![-919,217,1819,-755]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+5a^2-a-6)\) | = | \((-a)^{6}\cdot(-a^3+2a^2+2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 704 \) | = | \(2^{6}\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((32a^3-25a^2-115a+38)\) | = | \((-a)^{20}\cdot(-a^3+2a^2+2a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 11534336 \) | = | \(2^{20}\cdot11\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{85048962626261637}{44} a^{3} + \frac{115966996128885001}{44} a^{2} - \frac{33248386401040127}{22} a - \frac{72060916678769347}{44} \) | ||
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: | $\left(-\frac{7}{2} a^{3} + \frac{13}{2} a^{2} + \frac{17}{4} a - \frac{13}{2} : \frac{27}{8} a^{3} - \frac{25}{8} a^{2} - \frac{39}{8} a + \frac{15}{4} : 1\right)$ |
| 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: | \( 32.344159763435771309099027528851189699 \) | ||
| Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.45509286962894 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(4\) | \(I_{10}^{*}\) | Additive | \(1\) | \(6\) | \(20\) | \(2\) |
| \((-a^3+2a^2+2a-1)\) | \(11\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
704.3-r
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.