Base field \(\Q(\sqrt{65}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 16 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-16, -1, 1]))
gp: K = nfinit(Polrev([-16, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-22104,4876]),K([2538732,-560288])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-22104,4876]),Polrev([2538732,-560288])], K);
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-22104,4876],K![2538732,-560288]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((26,2a+12)\) | = | \((2,a)\cdot(2,a+1)\cdot(13,a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 52 \) | = | \(2\cdot2\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-28288a-26624)\) | = | \((2,a)^{19}\cdot(2,a+1)^{7}\cdot(13,a+6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 11341398016 \) | = | \(2^{19}\cdot2^{7}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((1664)\) | = | \((2,a)^{7}\cdot(2,a+1)^{7}\cdot(13,a+6)^{2}\) |
Minimal discriminant norm: | \( 2768896 \) | = | \(2^{7}\cdot2^{7}\cdot13^{2}\) |
j-invariant: | \( -\frac{2146689}{1664} \) | ||
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);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-26 a + 118 : 251 a - 1140 : 1\right)$ |
Height | \(0.19763543254554395156660112285556285713\) |
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: | \( 0.19763543254554395156660112285556285713 \) | ||
Period: | \( 18.894300301688451355019275666224417002 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot1\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.8526736948134195968379911319286008207 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
\((2,a+1)\) | \(2\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
\((13,a+6)\) | \(13\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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.6.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
52.1-h
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 338.a2 |
\(\Q\) | 650.g2 |