Properties

Label 2.0.8.1-19208.1-d1
Base field \(\Q(\sqrt{-2}) \)
Conductor \((98a)\)
Conductor norm \( 19208 \)
CM no
Base change yes: 3136.p4,392.d4
Q-curve yes
Torsion order \( 4 \)
Rank \( 2 \)

Related objects

Downloads

Learn more

Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-2}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+14{x}+80\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([14,0]),K([80,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([14,0])),Pol(Vecrev([80,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![14,0],K![80,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((98a)\) = \((a)^{3}\cdot(7)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 19208 \) = \(2^{3}\cdot49^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3294172)\) = \((a)^{4}\cdot(7)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10851569165584 \) = \(2^{4}\cdot49^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{432}{7} \)
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: \(2\)
Generators $\left(-4 : 7 a : 1\right)$ $\left(-\frac{147}{16} a + \frac{121}{32} : \frac{7179}{256} a + \frac{1225}{128} : 1\right)$
Heights \(1.89195900290132\) \(3.82501623624111\)
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: $\left(-10 : -20 a : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 3.65726503574073 \)
Period: \( 1.14751306237568 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.93511406302743 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)
\((7)\) \(49\) \(4\) \(I_1^{*}\) Additive \(1\) \(2\) \(7\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 19208.1-d consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of elliptic curves 3136.p4, 392.d4, defined over \(\Q\), so it is also a \(\Q\)-curve.