Elliptic curve 33930.8-d1 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-33930.8-d1
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $33930$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $7$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+i{x}{y}={x}^{3}+\left(174i-203\right){x}-1572i+507$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{7}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-3 i + \frac{19}{2} : \frac{53}{4} i - \frac{15}{4} : 1\right)$	$0.44024697283587001505713530219834350046$	$\infty$
$\left(-3 i - 13 : 47 i - 60 : 1\right)$	$0$	$7$

Invariants

Conductor:	$\frak{N}$	=	$(51i-177)$	=	$(i+1)\cdot(2i+1)\cdot(3)\cdot(2i+3)\cdot(2i+5)$
Conductor norm:	$N(\frak{N})$	=	$33930$	=	$2\cdot5\cdot9\cdot13\cdot29$
Discriminant:	$\Delta$	=	$-380345544i+299584008$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-380345544i+299584008)$	=	$(i+1)^{7}\cdot(2i+1)^{7}\cdot(3)^{7}\cdot(2i+3)^{2}\cdot(2i+5)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$234413310690000000$	=	$2^{7}\cdot5^{7}\cdot9^{7}\cdot13^{2}\cdot29$
j-invariant:	$j$	=	$-\frac{54981962133268901}{13398108750000} i + \frac{19988594138170643}{13398108750000}$
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:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.44024697283587001505713530219834350046$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.880493945671740030114270604396687000920$
Global period:	$\Omega(E/K)$	≈	$0.931112611681935644952419035286806698160$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$686$  =  $7\cdot7\cdot7\cdot2\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$7$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$5.7388731212718235054989276261861812125$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 5.738873121 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 0.931113 \cdot 0.880494 \cdot 686 } { {7^2 \cdot 2.000000} } \approx 5.738873121$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 5 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$7$	$I_{7}$	Split multiplicative	$-1$	$1$	$7$	$7$
$(2i+1)$	$5$	$7$	$I_{7}$	Split multiplicative	$-1$	$1$	$7$	$7$
$(3)$	$9$	$7$	$I_{7}$	Split multiplicative	$-1$	$1$	$7$	$7$
$(2i+3)$	$13$	$2$	$I_{2}$	Non-split multiplicative	$1$	$1$	$2$	$2$
$(2i+5)$	$29$	$1$	$I_{1}$	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 33930.8-d consists of curves linked by isogenies of degree 7.

Base change

This elliptic curve is not a $\Q$-curve.

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