Properties

Label 5.5.38569.1-11.1-b2
Base field 5.5.38569.1
Conductor norm \( 11 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 5.5.38569.1

Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
 
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(1965a^{4}+1435a^{3}-8543a^{2}-6367a+2230\right){x}+9983a^{4}+6859a^{3}-43325a^{2}-30702a+11057\)
sage: E = EllipticCurve([K([2,0,-4,0,1]),K([-3,1,1,0,0]),K([2,-3,-4,1,1]),K([2230,-6367,-8543,1435,1965]),K([11057,-30702,-43325,6859,9983])])
 
gp: E = ellinit([Polrev([2,0,-4,0,1]),Polrev([-3,1,1,0,0]),Polrev([2,-3,-4,1,1]),Polrev([2230,-6367,-8543,1435,1965]),Polrev([11057,-30702,-43325,6859,9983])], K);
 
magma: E := EllipticCurve([K![2,0,-4,0,1],K![-3,1,1,0,0],K![2,-3,-4,1,1],K![2230,-6367,-8543,1435,1965],K![11057,-30702,-43325,6859,9983]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+4a-2)\) = \((-a^3+a^2+4a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 11 \) = \(11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10004a^4-3192a^3-46419a^2+14331a+26476)\) = \((-a^3+a^2+4a-2)^{20}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -672749994932560009201 \) = \(-11^{20}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{38267658849058227209117423964971}{672749994932560009201} a^{4} - \frac{78660601240615996179672423975313}{672749994932560009201} a^{3} - \frac{45362244803600084463716093252952}{672749994932560009201} a^{2} + \frac{92081451737609143763002263964608}{672749994932560009201} a - \frac{20518519684505025230375543625252}{672749994932560009201} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
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);
 
Torsion generator: $\left(\frac{7}{4} a^{4} - \frac{1}{4} a^{3} - 4 a^{2} + \frac{15}{4} a - 6 : \frac{15}{8} a^{4} - \frac{23}{8} a^{3} - \frac{83}{8} a^{2} + \frac{47}{8} a + \frac{13}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.6125651092807530422844712223256707555 \)
Tamagawa product: \( 20 \)
Torsion order: \(2\)
Leading coefficient: \( 1.02637977 \)
Analytic order of Ш: \( 25 \) (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\((j)_{-}\)
\((-a^3+a^2+4a-2)\) \(11\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)

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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 11.1-b consists of curves linked by isogenies of degrees dividing 10.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.