Base field \(\Q(\sqrt{6}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
gp: K = nfinit(Polrev([-6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([-142,-61]),K([827,338])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,1]),Polrev([0,0]),Polrev([-142,-61]),Polrev([827,338])], K);
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,0],K![-142,-61],K![827,338]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((28)\) | = | \((-a+2)^{4}\cdot(7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 784 \) | = | \(2^{4}\cdot49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((43904a+87808)\) | = | \((-a+2)^{15}\cdot(7)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -3855122432 \) | = | \(-2^{15}\cdot49^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{4041775873}{1372} a + \frac{4951039391}{686} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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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 | $\left(-3 a - 4 : -12 a - 19 : 1\right)$ |
Height | \(0.059056996647282003071038008044511380920\) |
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: | \( 0.059056996647282003071038008044511380920 \) | ||
Period: | \( 7.4649903511912275912240599718918273847 \) | ||
Tamagawa product: | \( 12 \) = \(2^{2}\cdot3\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.1597636557957197494067942925987365838 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+2)\) | \(2\) | \(4\) | \(I_{7}^{*}\) | Additive | \(1\) | \(4\) | \(15\) | \(3\) |
\((7)\) | \(49\) | \(3\) | \(I_{3}\) | 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 |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
784.1-l
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.