Properties

Label 2.0.4.1-11250.3-f2
Base field \(\Q(\sqrt{-1}) \)
Conductor \((75i+75)\)
Conductor norm \( 11250 \)
CM no
Base change yes: 1200.g3,150.c3
Q-curve yes
Torsion order \( 10 \)
Rank \( 1 \)

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Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-1}) \)

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

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

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-28{x}+272\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-28,0]),K([272,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-28,0])),Pol(Vecrev([272,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-28,0],K![272,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((75i+75)\) = \((i+1)\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 11250 \) = \(2\cdot5^{2}\cdot5^{2}\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-31104000)\) = \((i+1)^{20}\cdot(-i-2)^{3}\cdot(2i+1)^{3}\cdot(3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 967458816000000 \) = \(2^{20}\cdot5^{3}\cdot5^{3}\cdot9^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{19465109}{248832} \)
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 $\left(4 i + 4 : 4 : 1\right)$
Height \(0.650527560569277\)
Torsion structure: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(32 : 164 : 1\right)$
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: \( 0.650527560569277 \)
Period: \( 0.787497134352966 \)
Tamagawa product: \( 400 \)  =  \(( 2^{2} \cdot 5 )\cdot2\cdot2\cdot5\)
Torsion order: \(10\)
Leading coefficient: \( 4.09830871812745 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)
\((-i-2)\) \(5\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)
\((2i+1)\) \(5\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)
\((3)\) \(9\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 11250.3-f consists of curves linked by isogenies of degrees dividing 10.

Base change

This curve is the base change of elliptic curves 1200.g3, 150.c3, defined over \(\Q\), so it is also a \(\Q\)-curve.