Base field 4.4.12400.1
Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 0, -12, 0, 1]))
gp: K = nfinit(Polrev([31, 0, -12, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-7/2,0,1/2]),K([7/2,-1,-1/2,0]),K([1,0,0,0]),K([-71/2,41/2,9/2,-5/2]),K([56,-57/2,-7,7/2])])
gp: E = ellinit([Polrev([0,-7/2,0,1/2]),Polrev([7/2,-1,-1/2,0]),Polrev([1,0,0,0]),Polrev([-71/2,41/2,9/2,-5/2]),Polrev([56,-57/2,-7,7/2])], K);
magma: E := EllipticCurve([K![0,-7/2,0,1/2],K![7/2,-1,-1/2,0],K![1,0,0,0],K![-71/2,41/2,9/2,-5/2],K![56,-57/2,-7,7/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/2a^2+a-1/2)\) | = | \((1/2a^2+a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1/2a^3+2a^2-3/2a-10)\) | = | \((1/2a^2+a-1/2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 81 \) | = | \(9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{29105995}{9} a^{3} + \frac{173474645}{18} a^{2} + \frac{36523615}{3} a - \frac{653017795}{18} \) | ||
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 | $\left(\frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{19}{10} a + \frac{39}{10} : -\frac{33}{50} a^{3} + \frac{26}{25} a^{2} + \frac{273}{50} a - \frac{266}{25} : 1\right)$ |
Height | \(0.30640035258027785447646698605488202880\) |
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: | $\left(-\frac{1}{2} a^{3} + a^{2} + \frac{9}{2} a - 9 : a^{3} - 2 a^{2} - 8 a + 15 : 1\right)$ |
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: | \( 0.30640035258027785447646698605488202880 \) | ||
Period: | \( 525.22095584861521324481889691911430445 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.89035015793608 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^2+a-1/2)\) | \(9\) | \(2\) | \(I_{2}\) | 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 |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.