Properties

Label 2.0.11.1-22275.8-a4
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 22275 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
 
gp: K = nfinit(Polrev([3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-533{x}-4598\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-533,0]),K([-4598,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-533,0]),Polrev([-4598,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-533,0],K![-4598,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-90a+45)\) = \((-a)^{2}\cdot(a-1)^{2}\cdot(-a-1)\cdot(a-2)\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 22275 \) = \(3^{2}\cdot3^{2}\cdot5\cdot5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5011875)\) = \((-a)^{6}\cdot(a-1)^{6}\cdot(-a-1)^{4}\cdot(a-2)^{4}\cdot(-2a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 25118891015625 \) = \(3^{6}\cdot3^{6}\cdot5^{4}\cdot5^{4}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22930509321}{6875} \)
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 $\left(-2 a - 14 : 8 a - 19 : 1\right)$ $\left(-\frac{113}{9} : -\frac{50}{27} a + \frac{181}{27} : 1\right)$
Heights \(0.32965794500566571076963965595464866547\) \(1.5067875968118718687009720926598132825\)
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: $\left(-\frac{53}{4} : \frac{49}{8} : 1\right)$
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: \( 0.49672450272502725455408875774082234875 \)
Period: \( 0.59020737031884682643386453810948630410 \)
Tamagawa product: \( 32 \)  =  \(2\cdot2\cdot2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 5.6572301021940543658904550048711050715 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((a-1)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((-a-1)\) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((a-2)\) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-2a+1)\) \(11\) \(2\) \(I_{2}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 22275.8-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 495.a1
\(\Q\) 5445.i1