Properties

Label 2.2.40.1-450.1-bh7
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 450 \)
CM no
Base change no
Q-curve yes
Torsion order \( 10 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{10}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, 0, 1]))
 
gp: K = nfinit(Polrev([-10, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}+\left(-1900a-7228\right){x}+92580a+292272\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-7228,-1900]),K([292272,92580])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-7228,-1900]),Polrev([292272,92580])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-7228,-1900],K![292272,92580]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((30,15a)\) = \((2,a)\cdot(3,a+1)\cdot(3,a+2)\cdot(5,a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 450 \) = \(2\cdot3\cdot3\cdot5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((213475500a+179091000)\) = \((2,a)^{5}\cdot(3,a+1)^{5}\cdot(3,a+2)^{20}\cdot(5,a)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 423644304721500000 \) = \(2^{5}\cdot3^{5}\cdot3^{20}\cdot5^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{309883187057876003779}{27894275208} a + \frac{122492090990437539176}{3486784401} \)
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 $\left(\frac{44}{9} a + \frac{458}{9} : -\frac{206}{27} a - \frac{242}{27} : 1\right)$
Height \(1.6302374790495324735237383037688429161\)
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: $\left(26 a + 62 : 212 a + 554 : 1\right)$
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: \( 1.6302374790495324735237383037688429161 \)
Period: \( 1.5651391437633482570940341859016168437 \)
Tamagawa product: \( 1000 \)  =  \(5\cdot5\cdot( 2^{2} \cdot 5 )\cdot2\)
Torsion order: \(10\)
Leading coefficient: \( 8.0687047953741175785598483750868177774 \)
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\((j)_{-}\)
\((2,a)\) \(2\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((3,a+1)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((3,a+2)\) \(3\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)
\((5,a)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

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, 4, 5, 10 and 20.
Its isogeny class 450.1-bh consists of curves linked by isogenies of degrees dividing 20.

Base change

This elliptic curve is a \(\Q\)-curve.

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