Elliptic curve 8100.2-b1 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-8100.2-b1
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $8100$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-190i+33\right){x}-571i+756$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(4 i - 6 : -13 i - 36 : 1\right)$	$1.2022257780883668684311453654863585916$	$\infty$
$\left(-\frac{19}{2} i - 6 : \frac{29}{4} i - \frac{9}{4} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(90)$	=	$(i+1)^{2}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}$
Conductor norm:	$N(\frak{N})$	=	$8100$	=	$2^{2}\cdot5\cdot5\cdot9^{2}$
Discriminant:	$\Delta$	=	$-55112400i+188956800$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-55112400i+188956800)$	=	$(i+1)^{8}\cdot(-i-2)^{2}\cdot(2i+1)^{6}\cdot(3)^{9}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$38742048900000000$	=	$2^{8}\cdot5^{2}\cdot5^{6}\cdot9^{9}$
j-invariant:	$j$	=	$-\frac{42660324}{15625} i + \frac{45166032}{15625}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.2022257780883668684311453654863585916$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$2.4044515561767337368622907309727171832$
Global period:	$\Omega(E/K)$	≈	$1.09058117626770329640249115511972163228$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$8$  =  $1\cdot2\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.6222496064139319502147628395855432672$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 2.622249606 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.090581 \cdot 2.404452 \cdot 8 } { {2^2 \cdot 2.000000} } \approx 2.622249606$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$1$	$IV^{*}$	Additive	$-1$	$2$	$8$	$0$
$(-i-2)$	$5$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$
$(2i+1)$	$5$	$2$	$I_{6}$	Non-split multiplicative	$1$	$1$	$6$	$6$
$(3)$	$9$	$2$	$III^{*}$	Additive	$1$	$2$	$9$	$0$

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
$3$	3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 8100.2-b consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is a $\Q$-curve.

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