Elliptic curve 4489.1-CMa1 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-4489.1-CMa1
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $4489$
• CM: yes ($-3$)
• Base change: no
• Q-curve: yes
• Torsion order: $1$
• Rank: not available

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

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

Weierstrass equation

${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+a{x}+2040a-987$

This is a global minimal model.

Mordell-Weil group structure

Not computed ($0 \le r \le 2$)

Invariants

Conductor:	$\frak{N}$	=	$(-77a+45)$	=	$(9a-7)^{2}$
Conductor norm:	$N(\frak{N})$	=	$4489$	=	$67^{2}$
Discriminant:	$\Delta$	=	$-58853475a+1378589443$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-58853475a+1378589443)$	=	$(9a-7)^{10}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$1822837804551761449$	=	$67^{10}$
j-invariant:	$j$	=	$0$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z[(1+\sqrt{-3})/2]$    (complex multiplication)
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-3})/2]$
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{U}(1)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r?$		$0 \le r \le 2$
Regulator*:	$\mathrm{Reg}(E/K)$	≈	$1$
Néron-Tate Regulator*:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1$
Global period:	$\Omega(E/K)$	≈	$0.844907201181166536160218623038423210180$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.9512296001687991653966573082374851614$
Analytic order of Ш*:	Ш${}_{\mathrm{an}}$	=	$4$ (rounded)

* Conditional on BSD: assuming rank = analytic rank.

Note: We expect that the nontriviality of Ш explains the discrepancy between the upper bound on the rank and the analytic rank. The application of further descents should suffice to establish the weak BSD conjecture for this curve.

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(9a-7)$	$67$	$1$	$II^{*}$	Additive	$-1$	$2$	$10$	$0$

Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p < 1000$ except those listed.

prime	Image of Galois Representation
$67$	67Cs.9.1

For all other primes $p$, the image is a Borel subgroup if $p=3$, a split Cartan subgroup if $\left(\frac{ -3 }{p}\right)=+1$ or a nonsplit Cartan subgroup if $\left(\frac{ -3 }{p}\right)=-1$.

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 4489.1-CMa consists of this curve only.

Base change

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

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