Base field 4.4.2225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-5/2,1/2,1/2]),K([4,7/2,-1/2,-1/2]),K([-2,-5/2,1/2,1/2]),K([38,93/2,-47/2,-35/2]),K([174,175,-141,-99])])
gp: E = ellinit([Polrev([-3,-5/2,1/2,1/2]),Polrev([4,7/2,-1/2,-1/2]),Polrev([-2,-5/2,1/2,1/2]),Polrev([38,93/2,-47/2,-35/2]),Polrev([174,175,-141,-99])], K);
magma: E := EllipticCurve([K![-3,-5/2,1/2,1/2],K![4,7/2,-1/2,-1/2],K![-2,-5/2,1/2,1/2],K![38,93/2,-47/2,-35/2],K![174,175,-141,-99]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1)\) | = | \((1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1 \) | = | 1 |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{11454772099655}{2} a^{3} + \frac{16476545963211}{2} a^{2} - \frac{17103076638427}{2} a - 9396387712126 \) | ||
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: | $\left(\frac{3}{2} a^{3} + \frac{3}{2} a^{2} - \frac{7}{2} a - 5 : \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{9}{2} a - 4 : 1\right)$ | $\left(\frac{3}{8} a^{3} - \frac{17}{8} a^{2} - \frac{17}{8} a + \frac{5}{2} : \frac{7}{8} a^{3} - 3 a - \frac{7}{8} : 1\right)$ |
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: | \( 30.765445936840999358367531220406381002 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 0.366877209043145 \) | ||
Analytic order of Ш: | \( 9 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
No primes of bad reduction.
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
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\(2\) | 2Cs |
\(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.