Properties

Label 5.5.70601.1-32.1-a1
Base field 5.5.70601.1
Conductor norm \( 32 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.70601.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+1\right){x}{y}+\left(2a^{4}-a^{3}-11a^{2}+a+6\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+4a\right){x}^{2}+\left(-3a^{3}+5a^{2}+9a-9\right){x}-3a^{4}+a^{3}+16a^{2}+4a-13\)
sage: E = EllipticCurve([K([1,2,-4,-1,1]),K([0,4,-3,-2,1]),K([6,1,-11,-1,2]),K([-9,9,5,-3,0]),K([-13,4,16,1,-3])])
 
gp: E = ellinit([Polrev([1,2,-4,-1,1]),Polrev([0,4,-3,-2,1]),Polrev([6,1,-11,-1,2]),Polrev([-9,9,5,-3,0]),Polrev([-13,4,16,1,-3])], K);
 
magma: E := EllipticCurve([K![1,2,-4,-1,1],K![0,4,-3,-2,1],K![6,1,-11,-1,2],K![-9,9,5,-3,0],K![-13,4,16,1,-3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 32 \) = \(32\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2)\) = \((2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32 \) = \(32\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2485641}{2} a^{4} + 1159797 a^{3} + 4981591 a^{2} - 2602560 a - 1253263 \)
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: \(1\)
Generator $\left(4 a^{4} - 3 a^{3} - 20 a^{2} + 2 a + 11 : -15 a^{4} + 10 a^{3} + 77 a^{2} - 4 a - 46 : 1\right)$
Height \(0.80165047343036683850476623374092870022\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.80165047343036683850476623374092870022 \)
Period: \( 176.91507373504903336389415777977262536 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.66879041 \)
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\((j)_{-}\)
\((2)\) \(32\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 32.1-a consists of this curve only.

Base change

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

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