Properties

Label 2.0.7.1-48400.8-a2
Base field \(\Q(\sqrt{-7}) \)
Conductor \((220)\)
Conductor norm \( 48400 \)
CM no
Base change yes: 10780.k4,220.a4
Q-curve yes
Torsion order \( 6 \)
Rank \( 2 \)

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

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

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

Weierstrass equation

\(y^2=x^{3}+x^{2}-45x+100\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-45,0]),K([100,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-45,0])),Pol(Vecrev([100,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-45,0],K![100,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((220)\) = \( \left(a\right)^{2} \cdot \left(-a + 1\right)^{2} \cdot \left(5\right) \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 48400 \) = \( 2^{4} \cdot 11^{2} \cdot 25 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((242000)\) = \( \left(a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(5\right)^{3} \cdot \left(-2 a + 3\right)^{2} \cdot \left(2 a + 1\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 58564000000 \) = \( 2^{8} \cdot 11^{4} \cdot 25^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{643956736}{15125} \)
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 $\left(-5 a - 5 : -5 a + 35 : 1\right)$ $\left(-5 : 15 : 1\right)$
Heights \(0.725194581699165\) \(0.731357122231623\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(15 : 55 : 1\right)$
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: \( 0.530376222329467 \)
Period: \( 1.43743301913193 \)
Tamagawa product: \( 108 \)  =  \(3\cdot3\cdot2\cdot2\cdot3\)
Torsion order: \(6\)
Leading coefficient: \(6.91566399019176\)
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(3\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\( \left(-a + 1\right) \) \(2\) \(3\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\( \left(-2 a + 3\right) \) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(2 a + 1\right) \) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(5\right) \) \(25\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 48400.8-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of elliptic curves 10780.k4, 220.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.