Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Invariants
| Conductor: | \((-a^5+4a^3-a^2-2a+2)\) | = | \((a^5-6a^3+a^2+9a-4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((7)\) | = | \((a^5-6a^3+a^2+9a-4)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 117649 \) | = | \(7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 16581375 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[\sqrt{-7}]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-a^{5} + 3 a^{4} + 7 a^{3} - 14 a^{2} - 11 a + 16 : -15 a^{5} + 4 a^{4} + 84 a^{3} - 41 a^{2} - 110 a + 70 : 1\right)$ |
| Height | \(0.21918026689575426058073252294022182761\) |
| Torsion structure: | \(\Z/14\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-2 a^{5} + 13 a^{3} - 2 a^{2} - 19 a + 4 : -4 a^{5} + 26 a^{3} - 4 a^{2} - 38 a + 6 : 1\right)$ |
| 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: | \( 0.21918026689575426058073252294022182761 \) | ||
| Period: | \( 125372.83396343342723708473628145604772 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(14\) | ||
| Leading coefficient: | \( 2.49749 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-6a^3+a^2+9a-4)\) | \(7\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
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.1.1[3] |
For all other primes \(p\), the image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 49.a1 |
| \(\Q\) | 441.c3 |
| \(\Q(\sqrt{21}) \) | 2.2.21.1-49.1-b3 |
| \(\Q(\zeta_{7})^+\) | a curve with conductor norm 35721 (not in the database) |
| \(\Q(\zeta_{7})^+\) | 3.3.49.1-49.1-a1 |