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([-5/2,1,1/2,0]),K([5/2,-5/2,-1/2,1/2]),K([1,-5/2,0,1/2]),K([14697/2,2762,-3973/2,-717]),K([-351115,-120844,93032,32235])])
gp: E = ellinit([Polrev([-5/2,1,1/2,0]),Polrev([5/2,-5/2,-1/2,1/2]),Polrev([1,-5/2,0,1/2]),Polrev([14697/2,2762,-3973/2,-717]),Polrev([-351115,-120844,93032,32235])], K);
magma: E := EllipticCurve([K![-5/2,1,1/2,0],K![5/2,-5/2,-1/2,1/2],K![1,-5/2,0,1/2],K![14697/2,2762,-3973/2,-717],K![-351115,-120844,93032,32235]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-1/2a^3+1/2a^2+7/2a-7/2)\) | = | \((-a+3)\cdot(-1/2a^3-3/2a^2+3/2a+11/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-6847a^3+16323a^2+17515a-14431)\) | = | \((-a+3)^{3}\cdot(-1/2a^3-3/2a^2+3/2a+11/2)^{24}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3814697265625000000 \) | = | \(4^{3}\cdot5^{24}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1455229050566292019}{1953125000} a^{3} - \frac{2823809491238563383}{1953125000} a^{2} - \frac{11985361149037390153}{1953125000} a + \frac{23257104737525574349}{1953125000} \) | ||
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 | $\left(-\frac{11}{2} a^{3} - 20 a^{2} + \frac{47}{2} a + 106 : -25 a^{3} - 65 a^{2} + 50 a + 90 : 1\right)$ |
Height | \(1.0079633739850880492272971638782303596\) |
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: | $\left(-\frac{27}{4} a^{3} - \frac{127}{8} a^{2} + \frac{111}{4} a + \frac{431}{8} : \frac{201}{16} a^{3} + \frac{655}{16} a^{2} - \frac{693}{16} a - \frac{2573}{16} : 1\right)$ |
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: | \( 1.0079633739850880492272971638782303596 \) | ||
Period: | \( 11.420599020407286331551508549226395712 \) | ||
Tamagawa product: | \( 24 \) = \(1\cdot( 2^{3} \cdot 3 )\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.48104153222950 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+3)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((-1/2a^3-3/2a^2+3/2a+11/2)\) | \(5\) | \(24\) | \(I_{24}\) | Split multiplicative | \(-1\) | \(1\) | \(24\) | \(24\) |
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
20.2-d
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.