Properties

Label 5.5.38569.1-17.1-c4
Base field 5.5.38569.1
Conductor norm \( 17 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+\left(-a^{4}+5a^{2}+a-3\right){x}{y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a\right){x}^{2}+\left(72a^{4}+36a^{3}-313a^{2}-173a+70\right){x}-339a^{4}-336a^{3}+1494a^{2}+1418a-496\)
sage: E = EllipticCurve([K([-3,1,5,0,-1]),K([0,-4,4,1,-1]),K([0,0,0,0,0]),K([70,-173,-313,36,72]),K([-496,1418,1494,-336,-339])])
 
gp: E = ellinit([Polrev([-3,1,5,0,-1]),Polrev([0,-4,4,1,-1]),Polrev([0,0,0,0,0]),Polrev([70,-173,-313,36,72]),Polrev([-496,1418,1494,-336,-339])], K);
 
magma: E := EllipticCurve([K![-3,1,5,0,-1],K![0,-4,4,1,-1],K![0,0,0,0,0],K![70,-173,-313,36,72],K![-496,1418,1494,-336,-339]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-3a-1)\) = \((a^3-3a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 17 \) = \(17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((22a^4+61a^3-130a^2-191a+150)\) = \((a^3-3a-1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6975757441 \) = \(17^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{63035509038380061495748094}{6975757441} a^{4} + \frac{128497588344189495041715327}{6975757441} a^{3} - \frac{53235816441571130890411893}{6975757441} a^{2} - \frac{108520961136492735262181368}{6975757441} a + \frac{30922567893134525558722397}{6975757441} \)
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 $\left(-\frac{124168}{32041} a^{4} - \frac{56131}{32041} a^{3} + \frac{496461}{32041} a^{2} + \frac{180140}{32041} a + \frac{65103}{32041} : \frac{21429659}{5735339} a^{4} + \frac{63107445}{5735339} a^{3} - \frac{86190463}{5735339} a^{2} - \frac{279886281}{5735339} a + \frac{52422108}{5735339} : 1\right)$
Height \(2.6128105177820355837973489989957110106\)
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);
 
Torsion generators: $\left(6 a^{4} + 2 a^{3} - 26 a^{2} - 10 a + 4 : -2 a^{4} + 9 a^{2} + 2 a - 2 : 1\right)$ $\left(-2 a^{4} - 2 a^{3} + 10 a^{2} + 10 a - 4 : -a^{2} - 2 a : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.6128105177820355837973489989957110106 \)
Period: \( 192.01825934281905933148811662306894564 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 1.59665569 \)
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\((j)_{-}\)
\((a^3-3a-1)\) \(17\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 17.1-c consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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