Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0,0,0]),K([-5,-19,-2,13,1,-2]),K([-2,-7,-3,6,1,-1]),K([3286,16027,1820,-10369,-743,1531]),K([72382,350175,38407,-226728,-15713,33441])])
gp: E = ellinit([Polrev([-3,0,1,0,0,0]),Polrev([-5,-19,-2,13,1,-2]),Polrev([-2,-7,-3,6,1,-1]),Polrev([3286,16027,1820,-10369,-743,1531]),Polrev([72382,350175,38407,-226728,-15713,33441])], K);
magma: E := EllipticCurve([K![-3,0,1,0,0,0],K![-5,-19,-2,13,1,-2],K![-2,-7,-3,6,1,-1],K![3286,16027,1820,-10369,-743,1531],K![72382,350175,38407,-226728,-15713,33441]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-5a^2+3a+4)\) | = | \((a^4-a^3-5a^2+3a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((29a^5-12a^4-194a^3+25a^2+280a+18)\) | = | \((a^4-a^3-5a^2+3a+4)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 17249876309 \) | = | \(29^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{13815841607213083975511386207}{17249876309} a^{5} - \frac{28062038812038612265502450888}{17249876309} a^{4} - \frac{39710623624428834678366798314}{17249876309} a^{3} + \frac{53026981056198164144695142492}{17249876309} a^{2} + \frac{44263596385560929969398311274}{17249876309} a + \frac{6801758503148238763013009216}{17249876309} \) | ||
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 | $\left(-\frac{158}{5} a^{5} + \frac{78}{5} a^{4} + 211 a^{3} - \frac{196}{5} a^{2} - \frac{1627}{5} a - 61 : -\frac{2818}{25} a^{5} + \frac{1404}{25} a^{4} + \frac{18974}{25} a^{3} - \frac{3496}{25} a^{2} - \frac{5867}{5} a - \frac{5676}{25} : 1\right)$ |
| Height | \(1.5450107495110274619548874675968178359\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.5450107495110274619548874675968178359 \) | ||
| Period: | \( 0.89757985382018758100737416302720210239 \) | ||
| Tamagawa product: | \( 7 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.56178 \) | ||
| Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-a^3-5a^2+3a+4)\) | \(29\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
29.1-a
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.