Properties

Label 2.2.13.1-108.1-a3
Base field \(\Q(\sqrt{13}) \)
Conductor norm \( 108 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{13}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-3, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(82a-195\right){x}-239a+538\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,1]),K([-195,82]),K([538,-239])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-195,82])),Pol(Vecrev([538,-239]))], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,1],K![-195,82],K![538,-239]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-6a)\) = \((-a)^{2}\cdot(-a+1)\cdot(2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 108 \) = \(3^{2}\cdot3\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4015332a+1417176)\) = \((-a)^{16}\cdot(-a+1)^{10}\cdot(2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 40669853253264 \) = \(3^{16}\cdot3^{10}\cdot4^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{476379541}{236196} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(4 a - 13 : 4 a - 6 : 1\right)$ $\left(-\frac{5}{4} a + \frac{5}{4} : -\frac{1}{2} a + \frac{15}{8} : 1\right)$
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: \( 4.01039243734724 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.11228273596834 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(4\) \(I_{10}^{*}\) Additive \(-1\) \(2\) \(16\) \(10\)
\((-a+1)\) \(3\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((2)\) \(4\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 108.1-a consists of curves linked by isogenies of degrees dividing 20.

Base change

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

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