Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-5202.2-e5
Conductor \((51 i + 51)\)
Conductor norm \( 5202 \)
CM no
base-change yes: 102.c2,816.b2
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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

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

magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);
 
sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)
 
gp (2.8): K = nfinit(i^2 + 1);
 

Weierstrass equation

\( y^2 + x y = x^{3} - 1734 x - 27936 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -1734, -27936]),K);
 
sage: E = EllipticCurve(K, [1, 0, 0, -1734, -27936])
 
gp (2.8): E = ellinit([1, 0, 0, -1734, -27936],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((51 i + 51)\) = \( \left(i + 1\right) \cdot \left(3\right) \cdot \left(i + 4\right) \cdot \left(i - 4\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 5202 \) = \( 2 \cdot 9 \cdot 17^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((27060804)\) = \( \left(i + 1\right)^{4} \cdot \left(3\right)^{4} \cdot \left(i + 4\right)^{4} \cdot \left(i - 4\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 732287113126416 \) = \( 2^{4} \cdot 9^{4} \cdot 17^{8} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{576615941610337}{27060804} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): 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: \( 1 \)
magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(-21 i - 21 : 63 i + 183 : 1\right)$

Height: 0.9567454423721191

magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 0.956745442372

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/4\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generators: $\left(-\frac{57}{2} : -\frac{153}{4} i + \frac{57}{4} : 1\right)$,$\left(-24 : 12 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(i + 4\right) \) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(i - 4\right) \) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 5202.2-e consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 102.c2, 816.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.