Properties

Label 2.2.8.1-2450.1-a5
Base field \(\Q(\sqrt{2}) \)
Conductor \((35a)\)
Conductor norm \( 2450 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(588a-3844\right){x}+20724a-93386\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([-3844,588]),K([-93386,20724])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-3844,588])),Pol(Vecrev([-93386,20724]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![-3844,588],K![-93386,20724]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((35a)\) = \((a)\cdot(-2a+1)\cdot(2a+1)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2450 \) = \(2\cdot7\cdot7\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((42973742000a+6770416800)\) = \((a)^{9}\cdot(-2a+1)\cdot(2a+1)^{18}\cdot(5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3647646459319405760000 \) = \(-2^{9}\cdot7\cdot7^{18}\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{136655602923200722901307}{1302730878328359200} a + \frac{73203665356140842512049}{651365439164179600} \)
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 $\left(-\frac{21}{2} a + \frac{239}{2} : \frac{583}{4} a - 1139 : 1\right)$
Height \(1.83715192182630\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-39 a + \frac{99}{4} : 19 a - \frac{99}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.83715192182630 \)
Period: \( 0.185753360121366 \)
Tamagawa product: \( 36 \)  =  \(1\cdot1\cdot( 2 \cdot 3^{2} )\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.17174715651869 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((-2a+1)\) \(7\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2a+1)\) \(7\) \(18\) \(I_{18}\) Split multiplicative \(-1\) \(1\) \(18\) \(18\)
\((5)\) \(25\) \(2\) \(I_{2}\) 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
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 2450.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

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