Elliptic curve 123904.1-h2 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-123904.1-h2
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $123904$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{-3})$

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

Weierstrass equation

${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+3\right){x}+3a$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(0 : a + 1 : 1\right)$	$0.73499280783000833513027721821155691386$	$\infty$
$\left(-a : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(352)$	=	$(2)^{5}\cdot(11)$
Conductor norm:	$N(\frak{N})$	=	$123904$	=	$4^{5}\cdot121$
Discriminant:	$\Delta$	=	$-704$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-704)$	=	$(2)^{6}\cdot(11)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$495616$	=	$4^{6}\cdot121$
j-invariant:	$j$	=	$\frac{46656}{11}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.73499280783000833513027721821155691386$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.46998561566001667026055443642311382772$
Global period:	$\Omega(E/K)$	≈	$8.8203984260625762963444875990081730764$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$2$  =  $2\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.7429210373168091095771689739145011583$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}3.742921037 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.820398 \cdot 1.469986 \cdot 2 } { {2^2 \cdot 1.732051} } \\ & \approx 3.742921037 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2)$	$4$	$2$	$III$	Additive	$-1$	$5$	$6$	$0$
$(11)$	$121$	$1$	$I_{1}$	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.
Its isogeny class 123904.1-h consists of curves linked by isogenies of degree 2.

Base change

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

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