Base field 4.4.18625.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 9, -14, -1, 1]))
gp: K = nfinit(Polrev([41, 9, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([7,1,-1,0]),K([1,0,0,0]),K([806/3,1229/3,-19,-121/3]),K([-113659/3,-74980/3,3325,7238/3])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([7,1,-1,0]),Polrev([1,0,0,0]),Polrev([806/3,1229/3,-19,-121/3]),Polrev([-113659/3,-74980/3,3325,7238/3])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![7,1,-1,0],K![1,0,0,0],K![806/3,1229/3,-19,-121/3],K![-113659/3,-74980/3,3325,7238/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3+a^2-5a-21/2)\) | = | \((1/2a^3+a^2-5a-21/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1/6a^3+a^2+1/3a-25/6)\) | = | \((1/2a^3+a^2-5a-21/2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5 \) | = | \(-5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4425908224139}{15} a^{3} + \frac{3342835384389}{5} a^{2} - \frac{29209317541288}{15} a - \frac{55559399005201}{15} \) | ||
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 | $\left(-\frac{2025}{121} a^{3} - \frac{3092}{121} a^{2} + \frac{20978}{121} a + \frac{34962}{121} : -\frac{1168463}{3993} a^{3} - \frac{571116}{1331} a^{2} + \frac{12123847}{3993} a + \frac{19382491}{3993} : 1\right)$ |
| Height | \(1.1553729021592402239676690510519194730\) |
| 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: | $\left(-\frac{4}{3} a^{3} - 3 a^{2} + \frac{41}{3} a + \frac{401}{12} : \frac{2}{3} a^{3} + \frac{3}{2} a^{2} - \frac{41}{6} a - \frac{413}{24} : 1\right)$ |
| 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: | \( 1.1553729021592402239676690510519194730 \) | ||
| Period: | \( 264.64365428982290951818533583308906806 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.24045136886674 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3+a^2-5a-21/2)\) | \(5\) | \(1\) | \(I_{1}\) | 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
5.1-d
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.