Properties

Label 2.2.28.1-648.1-d6
Base field \(\Q(\sqrt{7}) \)
Conductor norm \( 648 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{7}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
 
gp: K = nfinit(Polrev([-7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-357a-1563\right){x}-9110a-30545\)
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([0,0]),K([-1563,-357]),K([-30545,-9110])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,0]),Polrev([-1563,-357]),Polrev([-30545,-9110])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![-1563,-357],K![-30545,-9110]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((18a+54)\) = \((a+3)^{3}\cdot(-a+2)^{2}\cdot(-a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 648 \) = \(2^{3}\cdot3^{2}\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((742880160a-1929388896)\) = \((a+3)^{11}\cdot(-a+2)^{22}\cdot(-a-2)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -140555012843280384 \) = \(-2^{11}\cdot3^{22}\cdot3^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{400787258265625}{43046721} a + \frac{1060093826933375}{43046721} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{23}{2} a - 1 : \frac{25}{4} a + \frac{163}{4} : 1\right)$
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: \( 0.31975853161670030228501658739083237824 \)
Tamagawa product: \( 8 \)  =  \(1\cdot2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.9337178382833672380336712570132635616 \)
Analytic order of Ш: \( 16 \) (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\((j)_{-}\)
\((a+3)\) \(2\) \(1\) \(II^{*}\) Additive \(1\) \(3\) \(11\) \(0\)
\((-a+2)\) \(3\) \(4\) \(I_{16}^{*}\) Additive \(-1\) \(2\) \(22\) \(16\)
\((-a-2)\) \(3\) \(2\) \(I_{1}^{*}\) 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, 4, 8 and 16.
Its isogeny class 648.1-d consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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