Properties

Label 4.4.19600.1-29.3-a1
Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{5}, \sqrt{7})\)

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 29 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 16, -15, -2, 1]))
 
gp: K = nfinit(Polrev([29, 16, -15, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 16, -15, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-\frac{3}{23}a^{3}+\frac{16}{23}a^{2}+\frac{30}{23}a-\frac{102}{23}\right){x}{y}+\left(-\frac{3}{23}a^{3}+\frac{16}{23}a^{2}+\frac{30}{23}a-\frac{125}{23}\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(\frac{118}{23}a^{3}-\frac{361}{23}a^{2}-\frac{1433}{23}a+\frac{3414}{23}\right){x}+\frac{594}{23}a^{3}-\frac{1811}{23}a^{2}-\frac{6998}{23}a+\frac{16677}{23}\)
sage: E = EllipticCurve([K([-102/23,30/23,16/23,-3/23]),K([1,-1,0,0]),K([-125/23,30/23,16/23,-3/23]),K([3414/23,-1433/23,-361/23,118/23]),K([16677/23,-6998/23,-1811/23,594/23])])
 
gp: E = ellinit([Polrev([-102/23,30/23,16/23,-3/23]),Polrev([1,-1,0,0]),Polrev([-125/23,30/23,16/23,-3/23]),Polrev([3414/23,-1433/23,-361/23,118/23]),Polrev([16677/23,-6998/23,-1811/23,594/23])], K);
 
magma: E := EllipticCurve([K![-102/23,30/23,16/23,-3/23],K![1,-1,0,0],K![-125/23,30/23,16/23,-3/23],K![3414/23,-1433/23,-361/23,118/23],K![16677/23,-6998/23,-1811/23,594/23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4/23a^3+6/23a^2+63/23a-44/23)\) = \((-4/23a^3+6/23a^2+63/23a-44/23)\)
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: \((-29/23a^3+9/23a^2+336/23a+95/23)\) = \((-4/23a^3+6/23a^2+63/23a-44/23)^{3}\)
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: \( \frac{26708755}{560947} a^{3} - \frac{78978730}{560947} a^{2} - \frac{312565105}{560947} a + \frac{20685109}{19343} \)
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 $\left(-\frac{20}{23} a^{3} + \frac{99}{23} a^{2} - \frac{7}{23} a - \frac{220}{23} : \frac{165}{23} a^{3} - \frac{903}{23} a^{2} + \frac{558}{23} a + \frac{1217}{23} : 1\right)$
Height \(0.061084943140693153056514957568922378352\)
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: \( 0.061084943140693153056514957568922378352 \)
Period: \( 501.70341450493858853440672853273254363 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 2.62684496130221 \)
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\((j)_{-}\)
\((-4/23a^3+6/23a^2+63/23a-44/23)\) \(29\) \(3\) \(I_{3}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 29.3-a consists of curves linked by isogenies of degree 3.

Base change

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

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