Properties

Label 4.4.10512.1-36.1-f1
Base field 4.4.10512.1
Conductor norm \( 36 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.10512.1

Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
 
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3412a^{3}+4266a^{2}+18598a-2902\right){x}+154536a^{3}-193381a^{2}-839438a+122653\)
sage: E = EllipticCurve([K([-1,-4,-1,1]),K([-1,1,0,0]),K([-1,-5,-1,1]),K([-2902,18598,4266,-3412]),K([122653,-839438,-193381,154536])])
 
gp: E = ellinit([Polrev([-1,-4,-1,1]),Polrev([-1,1,0,0]),Polrev([-1,-5,-1,1]),Polrev([-2902,18598,4266,-3412]),Polrev([122653,-839438,-193381,154536])], K);
 
magma: E := EllipticCurve([K![-1,-4,-1,1],K![-1,1,0,0],K![-1,-5,-1,1],K![-2902,18598,4266,-3412],K![122653,-839438,-193381,154536]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-5a+2)\) = \((a^3-a^2-5a)\cdot(a^3-a^2-5a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 36 \) = \(4\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((32a^3-32a^2-160a-128)\) = \((a^3-a^2-5a)^{11}\cdot(a^3-a^2-5a-1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 37748736 \) = \(4^{11}\cdot9\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{19790001340417}{192} a^{3} - \frac{19790001340417}{192} a^{2} - \frac{98950006702085}{192} a + \frac{4947198382079}{48} \)
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: $\left(\frac{41}{2} a^{3} - \frac{101}{4} a^{2} - \frac{219}{2} a + \frac{35}{4} : -\frac{5}{4} a^{3} + \frac{3}{8} a^{2} + \frac{85}{8} a - \frac{13}{2} : 1\right)$
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: \( 22.401159090245306904877304802790972384 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.96639301883400 \)
Analytic order of Ш: \( 36 \) (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\((j)_{-}\)
\((a^3-a^2-5a)\) \(4\) \(1\) \(I_{11}\) Non-split multiplicative \(1\) \(1\) \(11\) \(11\)
\((a^3-a^2-5a-1)\) \(9\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 36.1-f 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.