Base field 4.4.5125.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
gp: K = nfinit(Polrev([11, 7, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,0,1,0]),K([-4,-5,0,1]),K([-3,0,1,0]),K([-141,69,47,-22]),K([592,-181,-189,67])])
gp: E = ellinit([Polrev([-4,0,1,0]),Polrev([-4,-5,0,1]),Polrev([-3,0,1,0]),Polrev([-141,69,47,-22]),Polrev([592,-181,-189,67])], K);
magma: E := EllipticCurve([K![-4,0,1,0],K![-4,-5,0,1],K![-3,0,1,0],K![-141,69,47,-22],K![592,-181,-189,67]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-3a^2+4a+10)\) | = | \((-a^3+a^2+4a+1)\cdot(a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 55 \) | = | \(5\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-25a^3+125a^2-375)\) | = | \((-a^3+a^2+4a+1)^{8}\cdot(a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 47265625 \) | = | \(5^{8}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{196392867540579}{3025} a^{3} + \frac{27744964314013}{605} a^{2} - \frac{802919148292512}{3025} a - \frac{72567111214406}{275} \) | ||
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 | $\left(-2 a^{3} + 6 a^{2} - 2 : 10 a^{3} - 43 a^{2} + 11 a + 81 : 1\right)$ |
Height | \(0.047398503847179915065632073065345758340\) |
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: | $\left(\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{4} a + \frac{27}{4} : \frac{5}{8} a^{3} - \frac{29}{8} a^{2} - \frac{3}{2} a + \frac{49}{4} : 1\right)$ |
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: | \( 0.047398503847179915065632073065345758340 \) | ||
Period: | \( 856.40968924317011665992094676002475013 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.26808466293632 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3+a^2+4a+1)\) | \(5\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((a-1)\) | \(11\) | \(2\) | \(I_{2}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
55.2-e
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.