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Properties

Label 4.4.19525.1-29.2-d1
Base field 4.4.19525.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.19525.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 14 x^{2} + 15 x + 45 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([45, 15, -14, -2, 1]))
 
gp: K = nfinit(Polrev([45, 15, -14, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, 15, -14, -2, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(\frac{1}{3}a^{3}-\frac{10}{3}a-2\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+\frac{8}{3}a-2\right){x}^{2}+\left(\frac{20}{3}a^{3}-\frac{10}{3}a^{2}-\frac{202}{3}a-70\right){x}-\frac{56}{3}a^{3}-\frac{130}{3}a^{2}+304a+518\)
sage: E = EllipticCurve([K([1,0,0,0]),K([-2,8/3,2/3,-1/3]),K([-2,-10/3,0,1/3]),K([-70,-202/3,-10/3,20/3]),K([518,304,-130/3,-56/3])])
 
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-2,8/3,2/3,-1/3]),Polrev([-2,-10/3,0,1/3]),Polrev([-70,-202/3,-10/3,20/3]),Polrev([518,304,-130/3,-56/3])], K);
 
magma: E := EllipticCurve([K![1,0,0,0],K![-2,8/3,2/3,-1/3],K![-2,-10/3,0,1/3],K![-70,-202/3,-10/3,20/3],K![518,304,-130/3,-56/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^2+2/3a-2)\) = \((1/3a^2+2/3a-2)\)
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: \((-2/3a^3+7a^2+92/3a-13)\) = \((1/3a^2+2/3a-2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -20511149 \) = \(-29^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5180611509923}{20511149} a^{3} - \frac{55887977545640}{61533447} a^{2} - \frac{128084474803192}{61533447} a + \frac{145674343256078}{20511149} \)
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: \( 244.57898476082517756965334320309218407 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.75034483211903 \)
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\((j)_{-}\)
\((1/3a^2+2/3a-2)\) \(29\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 29.2-d 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.