Elliptic curve 338.2-b2 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-338.2-b2
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $338$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $7$
• Rank: $0$

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

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

Weierstrass equation

${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{7}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-1 : -2 : 1\right)$	$0$	$7$

Invariants

Conductor:	$\frak{N}$	=	$(13i+13)$	=	$(i+1)\cdot(-3i-2)\cdot(2i+3)$
Conductor norm:	$N(\frak{N})$	=	$338$	=	$2\cdot13\cdot13$
Discriminant:	$\Delta$	=	$-1664$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-1664)$	=	$(i+1)^{14}\cdot(-3i-2)\cdot(2i+3)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$2768896$	=	$2^{14}\cdot13\cdot13$
j-invariant:	$j$	=	$-\frac{2146689}{1664}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$7.8417990398626970909740354626865420542$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$14$  =  $( 2 \cdot 7 )\cdot1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$7$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.1202570056946710129962907803837917220$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}1.120257006 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.841799 \cdot 1 \cdot 14 } { {7^2 \cdot 2.000000} } \\ & \approx 1.120257006 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$14$	$I_{14}$	Split multiplicative	$-1$	$1$	$14$	$14$
$(-3i-2)$	$13$	$1$	$I_{1}$	Non-split multiplicative	$1$	$1$	$1$	$1$
$(2i+3)$	$13$	$1$	$I_{1}$	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
$7$	7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 338.2-b consists of curves linked by isogenies of degree 7.

Base change

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

Base field	Curve
$\Q$	26.b2
$\Q$	208.d2