Properties

Label 2.0.4.1-97344.2-e3
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 97344 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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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]);
 

Weierstrass equation

\({y}^2={x}^{3}-i{x}^{2}+624{x}+9486i\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([624,0]),K([0,9486])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([624,0]),Polrev([0,9486])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![624,0],K![0,9486]]);
 

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

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-15 i - 9 : 0 : 1\right)$$0$$2$
$\left(31 i : 0 : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((312)\) = \((i+1)^{6}\cdot(3)\cdot(-3i-2)\cdot(2i+3)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 97344 \) = \(2^{6}\cdot9\cdot13\cdot13\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: $\Delta$ = $25022177856$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((25022177856)\) = \((i+1)^{12}\cdot(3)^{4}\cdot(-3i-2)^{6}\cdot(2i+3)^{6}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 626109384657296756736 \) = \(2^{12}\cdot9^{4}\cdot13^{6}\cdot13^{6}\)
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));
 
j-invariant: $j$ = \( -\frac{420526439488}{390971529} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(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: $r_{\mathrm{an}}$= \( 0 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content 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: $\Omega(E/K)$ \( 0.501352930208176441715359716908669240180 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 64 \)  =  \(2^{2}\cdot2^{2}\cdot2\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 4.0108234416654115337228777352693539215 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 4 \) (rounded)

BSD formula

$$\begin{aligned}4.010823442 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 0.501353 \cdot 1 \cdot 64 } { {4^2 \cdot 2.000000} } \\ & \approx 4.010823442 \end{aligned}$$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((i+1)\) \(2\) \(4\) \(I_{2}^{*}\) Additive \(1\) \(6\) \(12\) \(0\)
\((3)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-3i-2)\) \(13\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((2i+3)\) \(13\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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.
Its isogeny class 97344.2-e consists of curves linked by isogenies of degrees dividing 4.

Base change

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

Base field Curve
\(\Q\) 1248.j2
\(\Q\) 1248.e2