Base field 3.3.316.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
gp: K = nfinit(Polrev([2, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1]),K([4,1,-1]),K([0,0,0]),K([94,-138,-105]),K([1059,-1676,-1246])])
gp: E = ellinit([Polrev([-3,1,1]),Polrev([4,1,-1]),Polrev([0,0,0]),Polrev([94,-138,-105]),Polrev([1059,-1676,-1246])], K);
magma: E := EllipticCurve([K![-3,1,1],K![4,1,-1],K![0,0,0],K![94,-138,-105],K![1059,-1676,-1246]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a^2-5)\) | = | \((-a+1)\cdot(a^2-a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 242 \) | = | \(2\cdot11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((7190a^2-5223a-15017)\) | = | \((-a+1)^{14}\cdot(a^2-a-1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3512055906304 \) | = | \(2^{14}\cdot11^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1196719965343}{1982464} a^{2} + \frac{318008627741}{991232} a + \frac{2546638361901}{991232} \) | ||
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 | $\left(-a^{2} - 4 a - 1 : 10 a^{2} + 9 a - 10 : 1\right)$ | |
| Height | \(0.33447197324945116921576176724356068144\) | |
| 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: | $\left(\frac{19}{4} a^{2} + \frac{9}{2} a - \frac{25}{4} : -\frac{17}{2} a^{2} - \frac{107}{8} a + \frac{37}{8} : 1\right)$ | $\left(-2 a^{2} - 2 a + 2 : 4 a^{2} + 6 a - 3 : 1\right)$ |
| 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: | \( 0.33447197324945116921576176724356068144 \) | ||
| Period: | \( 15.235042414369934954026970153058945020 \) | ||
| Tamagawa product: | \( 56 \) = \(( 2 \cdot 7 )\cdot2^{2}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.0098798375705510165960148062390052883 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(2\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
| \((a^2-a-1)\) | \(11\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
242.3-d
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.