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([1,1]),K([1,1]),K([0,0]),K([27,-8]),K([253,-12])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,0]),Polrev([27,-8]),Polrev([253,-12])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![27,-8],K![253,-12]]);
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: | \((400,2a+62)\) | = | \((2,a)\cdot(2,a+1)^{4}\cdot(5,a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 800 \) | = | \(2\cdot2^{4}\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((6315008a-18250752)\) | = | \((2,a)^{10}\cdot(2,a+1)^{25}\cdot(5,a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 536870912000000 \) | = | \(2^{10}\cdot2^{25}\cdot5^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((-130048a+64512)\) | = | \((2,a)^{10}\cdot(2,a+1)^{13}\cdot(5,a+1)^{6}\) |
Minimal discriminant norm: | \( 131072000000 \) | = | \(2^{10}\cdot2^{13}\cdot5^{6}\) |
j-invariant: | \( -\frac{85169}{1024} a + \frac{182505}{1024} \) | ||
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: | \(2\) | |
Generators | $\left(-5 : -a - 6 : 1\right)$ | $\left(-6 a + 9 : 24 a - 31 : 1\right)$ |
Heights | \(0.51430768925891888519668190684707727355\) | \(0.73134822794979571054136917875023850840\) |
Torsion structure: | \(\Z/2\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(-3 a + 2 : 2 a - 13 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.36568898892385405623671089133478186542 \) | ||
Period: | \( 1.6477649488799772929740894709104098633 \) | ||
Tamagawa product: | \( 16 \) = \(2\cdot2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.4631896545783083615244449744610497647 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
\((2,a+1)\) | \(2\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
\((5,a+1)\) | \(5\) | \(2\) | \(I_0^{*}\) | 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 |
---|---|
\(2\) | 2B |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
800.13-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.