Base field \(\Q(\sqrt{5}, \sqrt{13})\)
Generator \(a\), with minimal polynomial \( x^{4} - 9 x^{2} + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 0, -9, 0, 1]))
gp: K = nfinit(Polrev([4, 0, -9, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -9, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1/2,1/2,0]),K([1/2,11/4,0,-1/4]),K([0,1,0,0]),K([-36,-81/2,5,9/2]),K([-124,-393/2,29/2,23])])
gp: E = ellinit([Polrev([-1,-1/2,1/2,0]),Polrev([1/2,11/4,0,-1/4]),Polrev([0,1,0,0]),Polrev([-36,-81/2,5,9/2]),Polrev([-124,-393/2,29/2,23])], K);
magma: E := EllipticCurve([K![-1,-1/2,1/2,0],K![1/2,11/4,0,-1/4],K![0,1,0,0],K![-36,-81/2,5,9/2],K![-124,-393/2,29/2,23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a+2)\) | = | \((1/2a^3-9/2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 16 \) | = | \(4^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-a^2+4)\) | = | \((1/2a^3-9/2a+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 256 \) | = | \(4^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2492391}{2} a^{3} + \frac{6521955}{2} a^{2} + 1430550 a - 943918 \) | ||
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: | \(0\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(\frac{3}{4} a^{3} - \frac{25}{4} a - \frac{3}{2} : \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{3}{2} : 1\right)$ | $\left(\frac{5}{8} a^{3} + \frac{1}{8} a^{2} - \frac{21}{4} a - 2 : \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - 3 a - \frac{3}{2} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1201.9885952425990633028929354534816121 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.15575826465634 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/2a^3-9/2a+1)\) | \(4\) | \(1\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.3-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.