Properties

Base field \(\Q(\sqrt{23}) \)
Label 2.2.92.1-200.1-a4
Conductor \((-10 a + 50)\)
Conductor norm \( 200 \)
CM no
base-change yes: 80.a1,21160.e1
Q-curve yes
Torsion order \( 4 \)
Rank not available

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Base field \(\Q(\sqrt{23}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 23 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 23)
 
gp: K = nfinit(a^2 - 23);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y = x^{3} + a x^{2} + \left(-6414 a - 30760\right) x + 584948 a + 2805312 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312]),K);
 
sage: E = EllipticCurve(K, [a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312])
 
gp: E = ellinit([a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-10 a + 50)\) = \( \left(-a + 5\right)^{3} \cdot \left(5\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 200 \) = \( 2^{3} \cdot 25 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((80)\) = \( \left(-a + 5\right)^{8} \cdot \left(5\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 6400 \) = \( 2^{8} \cdot 25 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{132304644}{5} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(12 a + 58 : -84 a - 402 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a + 5\right) \) \(2\) \(4\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(5\right) \) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 200.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 80.a1, 21160.e1, defined over \(\Q\), so it is also a \(\Q\)-curve.