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

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

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx([1, 0, 1]))
 

Weierstrass equation

\({y}^2+\left(i+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-4i+8\right){x}+8i-6\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,1]),K([-1,0]),K([0,0]),K([8,-4]),K([-6,8])])
 
Copy content gp:E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([8,-4]),Polrev([-6,8])], K);
 
Copy content magma:E := EllipticCurve([K![1,1],K![-1,0],K![0,0],K![8,-4],K![-6,8]]);
 
Copy content oscar:E = elliptic_curve([K([1,1]),K([-1,0]),K([0,0]),K([8,-4]),K([-6,8])])
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content sage:E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-3 i + 1 : 6 i - 2 : 1\right)$$0.047842369237747538265079771523321201483$$\infty$
$\left(-\frac{1}{2} i + 1 : -\frac{1}{4} i - \frac{3}{4} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((10i+80)\) = \((i+1)^{2}\cdot(-i-2)\cdot(2i+1)^{2}\cdot(2i+3)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Conductor norm: $N(\frak{N})$ = \( 6500 \) = \(2^{2}\cdot5\cdot5^{2}\cdot13\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(K, ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Copy content oscar:norm(conductor(E))
 
Discriminant: $\Delta$ = $58000i-4000$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((58000i-4000)\) = \((i+1)^{8}\cdot(-i-2)^{4}\cdot(2i+1)^{3}\cdot(2i+3)^{2}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 3380000000 \) = \(2^{8}\cdot5^{4}\cdot5^{3}\cdot13^{2}\)
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
Copy content oscar:norm(discriminant(E))
 
j-invariant: $j$ = \( \frac{22196836}{105625} i + \frac{124665848}{105625} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 1 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ ≈ \( 0.047842369237747538265079771523321201483 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 0.09568473847549507653015954304664240297 \)
Global period: $\Omega(E/K)$≈ \( 4.431415873287516380645740511993058676 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 48 \)  =  \(3\cdot2^{2}\cdot2\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(2\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 2.5441132134700418001516440244250828077 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}2.544113213 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 4.431416 \cdot 0.095685 \cdot 48 } { {2^2 \cdot 2.000000} } \\ & \approx 2.544113213 \end{aligned}$$

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content magma:LocalInformation(E);
 

This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((i+1)\) \(2\) \(3\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((-i-2)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((2i+1)\) \(5\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)
\((2i+3)\) \(13\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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.
Its isogeny class 6500.6-b consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a \(\Q\)-curve.

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