Properties

Label 4.4.19600.1-25.1-c1
Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Conductor \((4/23a^3-6/23a^2-40/23a+21/23)\)
Conductor norm \( 25 \)
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(Pol(Vecrev([29, 16, -15, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 16, -15, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-\frac{1}{23}a^{3}+\frac{13}{23}a^{2}-\frac{13}{23}a-\frac{103}{23}\right){y}={x}^{3}+\left(-\frac{1}{23}a^{3}+\frac{13}{23}a^{2}-\frac{13}{23}a-\frac{103}{23}\right){x}^{2}+\left(-\frac{5}{23}a^{3}-\frac{4}{23}a^{2}+\frac{27}{23}a+\frac{106}{23}\right){x}-\frac{3}{23}a^{3}-\frac{7}{23}a^{2}+\frac{7}{23}a-\frac{79}{23}\)
sage: E = EllipticCurve([K([0,0,0,0]),K([-103/23,-13/23,13/23,-1/23]),K([-103/23,-13/23,13/23,-1/23]),K([106/23,27/23,-4/23,-5/23]),K([-79/23,7/23,-7/23,-3/23])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-103/23,-13/23,13/23,-1/23])),Pol(Vecrev([-103/23,-13/23,13/23,-1/23])),Pol(Vecrev([106/23,27/23,-4/23,-5/23])),Pol(Vecrev([-79/23,7/23,-7/23,-3/23]))], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![-103/23,-13/23,13/23,-1/23],K![-103/23,-13/23,13/23,-1/23],K![106/23,27/23,-4/23,-5/23],K![-79/23,7/23,-7/23,-3/23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4/23a^3-6/23a^2-40/23a+21/23)\) = \((4/23a^3-6/23a^2-40/23a+21/23)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \(25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-25)\) = \((4/23a^3-6/23a^2-40/23a+21/23)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 390625 \) = \(25^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4096}{25} a^{2} + \frac{4096}{25} a + \frac{8192}{25} \)
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 $\left(\frac{2}{23} a^{3} - \frac{3}{23} a^{2} + \frac{3}{23} a + \frac{45}{23} : -\frac{4}{23} a^{3} - \frac{17}{23} a^{2} + \frac{40}{23} a + \frac{94}{23} : 1\right)$
Height \(0.147496153366697\)
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: \( 0.147496153366697 \)
Period: \( 255.852226101758 \)
Tamagawa product: \( 4 \)
Torsion order: \(1\)
Leading coefficient: \( 4.31282504917895 \)
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\((j)_{-}\)
\((4/23a^3-6/23a^2-40/23a+21/23)\) \(25\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 25.1-c consists of this curve only.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.