Properties

Label 2.2.28.1-126.1-f7
Base field \(\Q(\sqrt{7}) \)
Conductor \((-9 a + 21)\)
Conductor norm \( 126 \)
CM no
Base change yes: 42.a1,2352.l1
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 7)
 
gp: K = nfinit(a^2 - 7);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
 

Weierstrass equation

\(y^2+xy+y=x^{3}+x^{2}-1344x+18405\)
sage: E = EllipticCurve(K, [1, 1, 1, -1344, 18405])
 
gp: E = ellinit([1, 1, 1, -1344, 18405],K)
 
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -1344, 18405]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-9 a + 21)\) = \( \left(a + 3\right) \cdot \left(-a + 2\right) \cdot \left(-a - 2\right) \cdot \left(a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 126 \) = \( 2 \cdot 3^{2} \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((252)\) = \( \left(a + 3\right)^{4} \cdot \left(-a + 2\right)^{2} \cdot \left(-a - 2\right)^{2} \cdot \left(a\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 63504 \) = \( 2^{4} \cdot 3^{4} \cdot 7^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{268498407453697}{252} \)
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 $\left(\frac{5}{9} a + \frac{200}{9} : \frac{115}{27} a + \frac{10}{27} : 1\right)$
Height \(1.15540590735346\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{41}{2} : -\frac{3}{4} a - \frac{43}{4} : 1\right)$ $\left(12 a - 11 : -6 a + 5 : 1\right)$
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: \( 1.15540590735346 \)
Period: \( 12.0787350298211 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2\cdot2\cdot2\)
Torsion order: \(8\)
Leading coefficient: \(2.63740619695601\)
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\((j)_{-}\)
\( \left(a + 3\right) \) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-a + 2\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-a - 2\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a\right) \) \(7\) \(2\) \(I_{2}\) 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, 4 and 8.
Its isogeny class 126.1-f consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of elliptic curves 42.a1, 2352.l1, defined over \(\Q\), so it is also a \(\Q\)-curve.