Properties

Label 4.4.19225.1-29.1-a1
Base field 4.4.19225.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.19225.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 15 x^{2} + 2 x + 44 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([44, 2, -15, -1, 1]))
 
gp: K = nfinit(Polrev([44, 2, -15, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 2, -15, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+8\right){x}{y}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{7}{2}a-14\right){y}={x}^{3}+\left(\frac{3}{2}a^{3}-\frac{11}{2}a^{2}-\frac{21}{2}a+32\right){x}^{2}+\left(-59a^{3}+194a^{2}+433a-1112\right){x}+406a^{3}-1360a^{2}-2899a+7616\)
sage: E = EllipticCurve([K([8,-5/2,-3/2,1/2]),K([32,-21/2,-11/2,3/2]),K([-14,7/2,5/2,-1/2]),K([-1112,433,194,-59]),K([7616,-2899,-1360,406])])
 
gp: E = ellinit([Polrev([8,-5/2,-3/2,1/2]),Polrev([32,-21/2,-11/2,3/2]),Polrev([-14,7/2,5/2,-1/2]),Polrev([-1112,433,194,-59]),Polrev([7616,-2899,-1360,406])], K);
 
magma: E := EllipticCurve([K![8,-5/2,-3/2,1/2],K![32,-21/2,-11/2,3/2],K![-14,7/2,5/2,-1/2],K![-1112,433,194,-59],K![7616,-2899,-1360,406]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a+1)\) = \((a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((27a^3-95a^2-169a+475)\) = \((a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -24389 \) = \(-29^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{66138876736}{24389} a^{3} - \frac{59701564825}{24389} a^{2} + \frac{880560775027}{24389} a + \frac{1539517303357}{24389} \)
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 $\left(16 a^{3} - 53 a^{2} - 117 a + 302 : 164 a^{3} - 547 a^{2} - 1186 a + 3098 : 1\right)$
Height \(0.38564899147636837026120357769596824229\)
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: \( 0.38564899147636837026120357769596824229 \)
Period: \( 71.872031994433629459578194021091019630 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 2.39883399802785 \)
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\((j)_{-}\)
\((a+1)\) \(29\) \(3\) \(I_{3}\) 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 29.1-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.