Base field 4.4.17725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 13, -12, -2, 1]))
gp: K = nfinit(Polrev([41, 13, -12, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-7,-7,0,1]),K([6,7,0,-1]),K([0,0,0,0]),K([10895,-1553,-2703,658]),K([-370274,40801,89053,-20424])])
gp: E = ellinit([Polrev([-7,-7,0,1]),Polrev([6,7,0,-1]),Polrev([0,0,0,0]),Polrev([10895,-1553,-2703,658]),Polrev([-370274,40801,89053,-20424])], K);
magma: E := EllipticCurve([K![-7,-7,0,1],K![6,7,0,-1],K![0,0,0,0],K![10895,-1553,-2703,658],K![-370274,40801,89053,-20424]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-8a-13)\) | = | \((a^3+a^2-8a-13)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^3-18a+2)\) | = | \((a^3+a^2-8a-13)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 117649 \) | = | \(49^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1305047204432928684048645}{343} a^{3} - \frac{5054677974181230854600448}{343} a^{2} - \frac{6192263190544780991205150}{343} a + \frac{28564814649204996967288782}{343} \) | ||
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 | $\left(-3 a^{3} + 20 a^{2} - 4 a - 94 : -14 a^{3} + 66 a^{2} + 20 a - 290 : 1\right)$ |
| Height | \(1.0779552952515073919864183042768190239\) |
| 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: | $\left(-\frac{13}{4} a^{3} + \frac{37}{2} a^{2} - \frac{1}{2} a - \frac{343}{4} : -\frac{33}{8} a^{3} + \frac{247}{8} a^{2} - \frac{35}{4} a - \frac{303}{2} : 1\right)$ |
| 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: | \( 1.0779552952515073919864183042768190239 \) | ||
| Period: | \( 78.423891722548661923026639165345021522 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.90492233135250 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-8a-13)\) | \(49\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 and 4.
Its isogeny class
49.2-a
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.