Properties

Label 2.0.4.1-10816.2-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 10816 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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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(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(6i+3\right){x}-4i+5\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([1,1]),K([3,6]),K([5,-4])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([1,1]),Polrev([3,6]),Polrev([5,-4])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![1,1],K![3,6],K![5,-4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

trivial

Invariants

Conductor: $\frak{N}$ = \((104)\) = \((i+1)^{6}\cdot(-3i-2)\cdot(2i+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 10816 \) = \(2^{6}\cdot13\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $12480i+12376$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((12480i+12376)\) = \((i+1)^{6}\cdot(-3i-2)^{5}\cdot(2i+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 308915776 \) = \(2^{6}\cdot13^{5}\cdot13\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( -\frac{335147200}{371293} i + \frac{809589576}{371293} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(0\)
Regulator: $\mathrm{Reg}(E/K)$ = \( 1 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ = \( 1 \)
Global period: $\Omega(E/K)$ \( 5.3001384759435512371612959545701834004 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1 \)  =  \(1\cdot1\cdot1\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.6500692379717756185806479772850917002 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.650069238 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 5.300138 \cdot 1 \cdot 1 } { {1^2 \cdot 2.000000} } \approx 2.650069238$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((i+1)\) \(2\) \(1\) \(II\) Additive \(-1\) \(6\) \(6\) \(0\)
\((-3i-2)\) \(13\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((2i+3)\) \(13\) \(1\) \(I_{1}\) Non-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
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 10816.2-b consists of curves linked by isogenies of degree 5.

Base change

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

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