Properties

Label 2.2.5.1-2025.1-a1
Base field \(\Q(\sqrt{5}) \)
Conductor norm \( 2025 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{5}) \)

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

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

Weierstrass equation

\({y}^2+{y}={x}^{3}-75{x}+256\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([-75,0]),K([256,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-75,0]),Polrev([256,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![-75,0],K![256,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((45)\) = \((-2\phi+1)^{2}\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2025 \) = \(5^{2}\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1366875)\) = \((-2\phi+1)^{8}\cdot(3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1868347265625 \) = \(5^{8}\cdot9^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{102400}{3} \)
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 $\left(-5 : 22 : 1\right)$
Height \(0.025588090451262578988395786323209806501\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.025588090451262578988395786323209806501 \)
Period: \( 7.2681786971188661271646602995125008975 \)
Tamagawa product: \( 12 \)  =  \(3\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 1.9961340974161131949995494333548255584 \)
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\((j)_{-}\)
\((-2\phi+1)\) \(5\) \(3\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((3)\) \(9\) \(4\) \(I_{1}^{*}\) Additive \(1\) \(2\) \(7\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 2025.1-a consists of curves linked by isogenies of degree 5.

Base change

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

Base field Curve
\(\Q\) 225.a1
\(\Q\) 225.e1