Base field 3.3.1849.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 14 x - 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -14, -1, 1]))
gp: K = nfinit(Polrev([-8, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([-6,-3/2,1/2]),K([0,1,0]),K([497,1182,-287]),K([23202,70559/2,-17957/2])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-6,-3/2,1/2]),Polrev([0,1,0]),Polrev([497,1182,-287]),Polrev([23202,70559/2,-17957/2])], K);
magma: E := EllipticCurve([K![1,0,0],K![-6,-3/2,1/2],K![0,1,0],K![497,1182,-287],K![23202,70559/2,-17957/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^2+3/2a+1)\) | = | \((3/2a^2-5/2a-19)\cdot(-a-3)\) |
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: | \((-41/2a^2+165/2a+49)\) | = | \((3/2a^2-5/2a-19)^{2}\cdot(-a-3)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16384 \) | = | \(2^{2}\cdot2^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{58751319}{4096} a^{2} - \frac{129965209}{2048} a - \frac{1693955}{512} \) | ||
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 | $\left(-\frac{29}{2} a^{2} + \frac{59}{2} a + 162 : \frac{285}{2} a^{2} - \frac{463}{2} a - 1850 : 1\right)$ | |
| Height | \(0.76731573100008332427238108557282013956\) | |
| 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: | $\left(3 a^{2} - 16 a + 12 : -\frac{3}{2} a^{2} + \frac{15}{2} a - 6 : 1\right)$ | $\left(-\frac{7}{4} a^{2} + 9 a - \frac{17}{4} : \frac{7}{8} a^{2} - 5 a + \frac{17}{8} : 1\right)$ |
| 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: | \( 0.76731573100008332427238108557282013956 \) | ||
| Period: | \( 77.318520343963846450351764636028289985 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.0347857608879087113849882983467792446 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3/2a^2-5/2a-19)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-a-3)\) | \(2\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
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 and 4.
Its isogeny class
4.3-b
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.