Properties

Label 4.4.19600.1-29.2-a2
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

\({y}^2+\left(-\frac{4}{23}a^{3}+\frac{6}{23}a^{2}+\frac{63}{23}a-\frac{21}{23}\right){x}{y}+\left(-\frac{2}{23}a^{3}+\frac{3}{23}a^{2}+\frac{43}{23}a+\frac{1}{23}\right){y}={x}^{3}+\left(-\frac{1}{23}a^{3}-\frac{10}{23}a^{2}+\frac{33}{23}a+\frac{104}{23}\right){x}^{2}+\left(\frac{17}{23}a^{3}-\frac{106}{23}a^{2}-\frac{147}{23}a+\frac{1153}{23}\right){x}-\frac{160}{23}a^{3}+\frac{424}{23}a^{2}+\frac{1922}{23}a-\frac{3577}{23}\)
sage: E = EllipticCurve([K([-21/23,63/23,6/23,-4/23]),K([104/23,33/23,-10/23,-1/23]),K([1/23,43/23,3/23,-2/23]),K([1153/23,-147/23,-106/23,17/23]),K([-3577/23,1922/23,424/23,-160/23])])
 
gp: E = ellinit([Polrev([-21/23,63/23,6/23,-4/23]),Polrev([104/23,33/23,-10/23,-1/23]),Polrev([1/23,43/23,3/23,-2/23]),Polrev([1153/23,-147/23,-106/23,17/23]),Polrev([-3577/23,1922/23,424/23,-160/23])], K);
 
magma: E := EllipticCurve([K![-21/23,63/23,6/23,-4/23],K![104/23,33/23,-10/23,-1/23],K![1/23,43/23,3/23,-2/23],K![1153/23,-147/23,-106/23,17/23],K![-3577/23,1922/23,424/23,-160/23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4/23a^3+6/23a^2+63/23a-21/23)\) = \((-4/23a^3+6/23a^2+63/23a-21/23)\)
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: \((-4/23a^3+6/23a^2+63/23a-21/23)\) = \((-4/23a^3+6/23a^2+63/23a-21/23)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29 \) = \(29\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2260418171}{667} a^{3} - \frac{11897954852}{667} a^{2} + \frac{4926111247}{667} a + \frac{692520183}{23} \)
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 $\left(\frac{6}{23} a^{3} - \frac{9}{23} a^{2} - \frac{83}{23} a + \frac{89}{23} : -\frac{13}{23} a^{3} + \frac{31}{23} a^{2} + \frac{153}{23} a - \frac{235}{23} : 1\right)$
Height \(0.18325482942207945916954487270676713506\)
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: \( 0.18325482942207945916954487270676713506 \)
Period: \( 501.70341450493858853440672853273254364 \)
Tamagawa product: \( 1 \)
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\((j)_{-}\)
\((-4/23a^3+6/23a^2+63/23a-21/23)\) \(29\) \(1\) \(I_{1}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 29.2-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.