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([4,-1/2,-1/2]),K([1,0,0]),K([-2806,-3935,1035]),K([-105534,-148183,38047])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([4,-1/2,-1/2]),Polrev([1,0,0]),Polrev([-2806,-3935,1035]),Polrev([-105534,-148183,38047])], K);
magma: E := EllipticCurve([K![1,0,0],K![4,-1/2,-1/2],K![1,0,0],K![-2806,-3935,1035],K![-105534,-148183,38047]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/2a^2-1/2a)\) | = | \((1/2a^2-1/2a-8)\cdot(-a-3)\) |
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: | \((-3/2a^2+5/2a+18)\) | = | \((1/2a^2-1/2a-8)^{3}\cdot(-a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 32 \) | = | \(2^{3}\cdot2^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{31087063737}{16} a^{2} + \frac{50207835865}{16} a + \frac{202168749849}{8} \) | ||
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 | $\left(-a^{2} + 5 a + 2 : 31 a^{2} - 126 a - 89 : 1\right)$ |
| Height | \(0.76731573100008332427238108557282013956\) |
| 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: | $\left(-\frac{15}{8} a^{2} + \frac{65}{8} a + \frac{17}{4} : \frac{15}{16} a^{2} - \frac{65}{16} a - \frac{21}{8} : 1\right)$ |
| 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: | \( 38.659260171981923225175882318014144993 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^2-1/2a-8)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((-a-3)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
4.2-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.