Properties

Label 5.5.157457.1-15.1-b1
Base field 5.5.157457.1
Conductor norm \( 15 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.157457.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-1, 4, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-a^{2}-2a\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+a+1\right){x}^{2}+\left(a^{3}-a^{2}+1\right){x}-20a^{4}+58a^{3}+33a^{2}-131a+23\)
sage: E = EllipticCurve([K([1,-3,-1,1,0]),K([1,1,-4,-1,1]),K([0,-2,-1,1,0]),K([1,0,-1,1,0]),K([23,-131,33,58,-20])])
 
gp: E = ellinit([Polrev([1,-3,-1,1,0]),Polrev([1,1,-4,-1,1]),Polrev([0,-2,-1,1,0]),Polrev([1,0,-1,1,0]),Polrev([23,-131,33,58,-20])], K);
 
magma: E := EllipticCurve([K![1,-3,-1,1,0],K![1,1,-4,-1,1],K![0,-2,-1,1,0],K![1,0,-1,1,0],K![23,-131,33,58,-20]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+2a^3+3a^2-3a-1)\) = \((a-1)\cdot(a^2-a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 15 \) = \(3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4+3a^3-2a-9)\) = \((a-1)^{6}\cdot(a^2-a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -91125 \) = \(-3^{6}\cdot5^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{815895463534687}{91125} a^{4} + \frac{571481037512924}{18225} a^{3} - \frac{1028729291749087}{91125} a^{2} - \frac{844716480359038}{30375} a + \frac{543144154205414}{91125} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 121.98084393238314947554561176305488811 \)
Tamagawa product: \( 6 \)  =  \(2\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 1.84442881 \)
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\((j)_{-}\)
\((a-1)\) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((a^2-a-2)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Ns

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 15.1-b 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.