Elliptic curve 52000.5-g4 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-52000.5-g4
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $52000$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+\left(i+1\right){x}{y}={x}^{3}-i{x}^{2}+\left(-202i-134\right){x}+1468i+101$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(4 i + 8 : -4 i - 3 : 1\right)$	$0.60363960279506303965725667597077428601$	$\infty$
$\left(\frac{9}{2} i + 8 : -\frac{25}{4} i - \frac{7}{4} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(60i-220)$	=	$(i+1)^{5}\cdot(-i-2)\cdot(2i+1)^{2}\cdot(-3i-2)$
Conductor norm:	$N(\frak{N})$	=	$52000$	=	$2^{5}\cdot5\cdot5^{2}\cdot13$
Discriminant:	$\Delta$	=	$-181800i-412400$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-181800i-412400)$	=	$(i+1)^{6}\cdot(-i-2)^{2}\cdot(2i+1)^{10}\cdot(-3i-2)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$203125000000$	=	$2^{6}\cdot5^{2}\cdot5^{10}\cdot13$
j-invariant:	$j$	=	$-\frac{16041806568}{8125} i + \frac{23413439024}{8125}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.60363960279506303965725667597077428601$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.20727920559012607931451335194154857202$
Global period:	$\Omega(E/K)$	≈	$1.75539284778371904727523794248764166290$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$16$  =  $2\cdot2\cdot2^{2}\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$4.2384985655418348845143022586582911597$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}4.238498566 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.755393 \cdot 1.207279 \cdot 16 } { {2^2 \cdot 2.000000} } \\ & \approx 4.238498566 \end{aligned}$

Local data at primes of bad reduction

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$2$	$III$	Additive	$1$	$5$	$6$	$0$
$(-i-2)$	$5$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$
$(2i+1)$	$5$	$4$	$I_{4}^{*}$	Additive	$1$	$2$	$10$	$4$
$(-3i-2)$	$13$	$1$	$I_{1}$	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
$2$	2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 52000.5-g consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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