Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, -9, -1, 1]))
gp: K = nfinit(Polrev([10, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,0,1]),K([-1,-1,0]),K([1,0,0]),K([770,-449,-230]),K([14613,-8453,-4360])])
gp: E = ellinit([Polrev([-5,0,1]),Polrev([-1,-1,0]),Polrev([1,0,0]),Polrev([770,-449,-230]),Polrev([14613,-8453,-4360])], K);
magma: E := EllipticCurve([K![-5,0,1],K![-1,-1,0],K![1,0,0],K![770,-449,-230],K![14613,-8453,-4360]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a)\) | = | \((2,a)\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((5a^2-3a-30)\) | = | \((2,a)^{2}\cdot(5,a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -2500 \) | = | \(-2^{2}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{153118770135195129}{2500} a^{2} + \frac{292341679554580081}{2500} a - \frac{525519586917853471}{2500} \) | ||
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 | $\left(-\frac{197}{50} a^{2} - \frac{34}{5} a + \frac{2471}{100} : \frac{3351}{500} a^{2} + \frac{3119}{500} a - \frac{25951}{1000} : 1\right)$ |
Height | \(5.5105057198376892053697448515480342368\) |
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: | $\left(-4 a^{2} - \frac{19}{4} a + \frac{77}{4} : \frac{11}{4} a^{2} + \frac{15}{2} a + \frac{31}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 5.5105057198376892053697448515480342368 \) | ||
Period: | \( 1.9098549873380146151666310342383169586 \) | ||
Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.7096262381248791008264610586228142042 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((5,a)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
10.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.