Properties

Label 4.4.18625.1-11.1-b2
Base field 4.4.18625.1
Conductor norm \( 11 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 4.4.18625.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 9, -14, -1, 1]))
 
gp: K = nfinit(Polrev([41, 9, -14, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{6}a^{3}+a^{2}-\frac{1}{3}a-\frac{35}{6}\right){x}{y}+\left(a^{2}+a-6\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(\frac{107}{6}a^{3}+34a^{2}-\frac{524}{3}a-\frac{2023}{6}\right){x}-\frac{337}{3}a^{3}-142a^{2}+\frac{3587}{3}a+\frac{5222}{3}\)
sage: E = EllipticCurve([K([-35/6,-1/3,1,1/6]),K([-1,-1,0,0]),K([-6,1,1,0]),K([-2023/6,-524/3,34,107/6]),K([5222/3,3587/3,-142,-337/3])])
 
gp: E = ellinit([Polrev([-35/6,-1/3,1,1/6]),Polrev([-1,-1,0,0]),Polrev([-6,1,1,0]),Polrev([-2023/6,-524/3,34,107/6]),Polrev([5222/3,3587/3,-142,-337/3])], K);
 
magma: E := EllipticCurve([K![-35/6,-1/3,1,1/6],K![-1,-1,0,0],K![-6,1,1,0],K![-2023/6,-524/3,34,107/6],K![5222/3,3587/3,-142,-337/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/6a^3-7/3a-11/6)\) = \((1/6a^3-7/3a-11/6)\)
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: \((-1/6a^3+10/3a-25/6)\) = \((1/6a^3-7/3a-11/6)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 121 \) = \(11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4823165761}{242} a^{3} - \frac{6386985120}{121} a^{2} - \frac{23238969544}{121} a + \frac{120028019549}{242} \)
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 $\left(\frac{1}{8} a^{3} - \frac{15}{4} a^{2} - \frac{3}{2} a + \frac{273}{8} : \frac{185}{24} a^{3} - \frac{109}{8} a^{2} - \frac{1711}{24} a + \frac{1633}{12} : 1\right)$
Height \(1.3050711245166679721433609244299533266\)
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: $\left(\frac{5}{8} a^{3} + \frac{5}{4} a^{2} - \frac{15}{2} a - \frac{139}{8} : \frac{1}{24} a^{3} + \frac{13}{8} a^{2} + \frac{91}{24} a - \frac{17}{6} : 1\right)$ $\left(-\frac{2}{3} a^{3} - 2 a^{2} + \frac{19}{3} a + \frac{55}{3} : 2 a^{3} + 2 a^{2} - 17 a - 13 : 1\right)$
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: \( 1.3050711245166679721433609244299533266 \)
Period: \( 756.70631293927004821000076957810414603 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 3.61812362218058 \)
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\((j)_{-}\)
\((1/6a^3-7/3a-11/6)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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.
Its isogeny class 11.1-b consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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