Properties

Label 2.0.11.1-225.3-b3
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 225 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8a-20\right){x}-18a+25\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([1,1]),K([-20,8]),K([25,-18])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([1,1]),Polrev([-20,8]),Polrev([25,-18])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![1,1],K![-20,8],K![25,-18]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((7a+6)\) = \((-a)^{2}\cdot(a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 225 \) = \(3^{2}\cdot5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1367a-437736)\) = \((-a)^{9}\cdot(a-2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 192216796875 \) = \(3^{9}\cdot5^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{24167}{27} a + \frac{28561}{27} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.5493087859339465648591909196425806871 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 1.8685367008523471230184362528253940988 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_{3}^{*}\) Additive \(-1\) \(2\) \(9\) \(3\)
\((a-2)\) \(5\) \(1\) \(II^{*}\) Additive \(1\) \(2\) \(10\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 225.3-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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

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