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([1,1,0,0]),K([5/2,1,-1/2,0]),K([-7/2,0,1/2,0]),K([26,3/2,-10,5/2]),K([-153,83/2,52,-33/2])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([5/2,1,-1/2,0]),Polrev([-7/2,0,1/2,0]),Polrev([26,3/2,-10,5/2]),Polrev([-153,83/2,52,-33/2])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![5/2,1,-1/2,0],K![-7/2,0,1/2,0],K![26,3/2,-10,5/2],K![-153,83/2,52,-33/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/2a^3-7/2a-1)\) | = | \((1/2a^3-7/2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1/2a^2+a+5/2)\) | = | \((1/2a^3-7/2a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -19 \) | = | \(-19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{7038892130560}{19} a^{3} - \frac{13656052294240}{19} a^{2} + \frac{57972794066560}{19} a + \frac{112472175052640}{19} \) | ||
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 | $\left(-a + 1 : -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} + \frac{5}{2} a - \frac{9}{2} : 1\right)$ |
Height | \(0.56410764919456171565791407607654792931\) |
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: | $\left(-\frac{1}{2} a^{3} + \frac{5}{4} a^{2} + 3 a - \frac{35}{4} : -\frac{3}{8} a^{3} + \frac{5}{8} a^{2} + \frac{23}{8} a - \frac{13}{8} : 1\right)$ |
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: | \( 0.56410764919456171565791407607654792931 \) | ||
Period: | \( 453.85755379731274289934979975150364768 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.29916804176374 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-7/2a-1)\) | \(19\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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
19.1-a
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.