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Properties

Label 4.4.3600.1-1.1-a1
Base field \(\Q(\sqrt{3}, \sqrt{5})\)
Conductor norm \( 1 \)
CM yes (\(-240\))
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{3}, \sqrt{5})\)

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 8, -7, -2, 1]))
 
gp: K = nfinit(Polrev([1, 8, -7, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -7, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-\frac{4}{7}a^{3}+\frac{6}{7}a^{2}+\frac{31}{7}a-\frac{20}{7}\right){x}{y}+\left(-\frac{4}{7}a^{3}+\frac{6}{7}a^{2}+\frac{31}{7}a-\frac{20}{7}\right){y}={x}^{3}+\left(\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{20}{7}a-\frac{16}{7}\right){x}^{2}+\left(-\frac{233}{7}a^{3}+\frac{472}{7}a^{2}+\frac{1678}{7}a-\frac{2166}{7}\right){x}+\frac{1595}{7}a^{3}-\frac{3502}{7}a^{2}-\frac{10207}{7}a+\frac{13582}{7}\)
sage: E = EllipticCurve([K([-20/7,31/7,6/7,-4/7]),K([-16/7,-20/7,2/7,1/7]),K([-20/7,31/7,6/7,-4/7]),K([-2166/7,1678/7,472/7,-233/7]),K([13582/7,-10207/7,-3502/7,1595/7])])
 
gp: E = ellinit([Polrev([-20/7,31/7,6/7,-4/7]),Polrev([-16/7,-20/7,2/7,1/7]),Polrev([-20/7,31/7,6/7,-4/7]),Polrev([-2166/7,1678/7,472/7,-233/7]),Polrev([13582/7,-10207/7,-3502/7,1595/7])], K);
 
magma: E := EllipticCurve([K![-20/7,31/7,6/7,-4/7],K![-16/7,-20/7,2/7,1/7],K![-20/7,31/7,6/7,-4/7],K![-2166/7,1678/7,472/7,-233/7],K![13582/7,-10207/7,-3502/7,1595/7]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -144059431778799259905 a^{3} + 127634233646510918145 a^{2} + 1150602705390880344330 a + 129315311353667976165 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-60}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1}{2} a^{2} + 2 a + 1 : \frac{75}{14} a^{3} - \frac{309}{28} a^{2} - \frac{1005}{28} a + \frac{1177}{28} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 558.15084293028806704677226950282572472 \)
Tamagawa product: \( 1 \)
Torsion order: \(4\)
Leading coefficient: \( 0.581407128052383 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

Galois Representations

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

The image is a Borel subgroup if \(p\in \{ 2, 3, 5\}\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 60.

Base change

This elliptic curve is a \(\Q\)-curve.

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