Base field \(\Q(\sqrt{-31}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 8 \); class number \(3\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -1, 1]))
gp: K = nfinit(Polrev([8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([2,1]),K([-3,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([2,1]),Polrev([-3,0])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![2,1],K![-3,0]]);
This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((8,4a+4)\) | = | \((2,a)^{2}\cdot(2,a+1)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 32 \) | = | \(2^{2}\cdot2^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((240a+656)\) | = | \((2,a)^{4}\cdot(2,a+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1048576 \) | = | \(2^{4}\cdot2^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((16)\) | = | \((2,a)^{4}\cdot(2,a+1)^{4}\) |
Minimal discriminant norm: | \( 256 \) | = | \(2^{4}\cdot2^{4}\) |
j-invariant: | \( -1536 a + 2816 \) | ||
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 | $\left(-1 : a - 1 : 1\right)$ |
Height | \(0.13915837929036190475801962283787388154\) |
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);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.13915837929036190475801962283787388154 \) | ||
Period: | \( 8.1916091217996210905920816139725770143 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.6379012830066939029335552834961976507 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
\((2,a+1)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(3\) | \(4\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cn |
\(5\) | 5S4 |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 32.4-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.