Properties

Label 2.2.8.1-882.1-f2
Base field \(\Q(\sqrt{2}) \)
Conductor norm \( 882 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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(Polrev([-2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-18a-27\right){x}+51a+72\)
sage: E = EllipticCurve([K([1,0]),K([0,-1]),K([1,0]),K([-27,-18]),K([72,51])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([1,0]),Polrev([-27,-18]),Polrev([72,51])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,-1],K![1,0],K![-27,-18],K![72,51]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((21a)\) = \((a)\cdot(-2a+1)\cdot(2a+1)\cdot(3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 882 \) = \(2\cdot7\cdot7\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4410a-882)\) = \((a)^{2}\cdot(-2a+1)^{2}\cdot(2a+1)^{4}\cdot(3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 38118276 \) = \(2^{2}\cdot7^{2}\cdot7^{4}\cdot9^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{384494749}{21609} a + \frac{150700299}{4802} \)
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 $\left(7 a + 10 : 25 a + 35 : 1\right)$
Height \(0.065225884120464496366333005040062132070\)
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: $\left(-2 a - 5 : a + 2 : 1\right)$ $\left(a + \frac{7}{4} : -\frac{1}{2} a - \frac{11}{8} : 1\right)$
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: \( 0.065225884120464496366333005040062132070 \)
Period: \( 17.236294500294271387168138531352251168 \)
Tamagawa product: \( 32 \)  =  \(2\cdot2\cdot2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.5899332005497986378925940077662732221 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2a+1)\) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3)\) \(9\) \(2\) \(I_{2}\) Non-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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 882.1-f consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a \(\Q\)-curve.

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