Elliptic curve 55552.2-e5 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-55552.2-e5
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $55552$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $1$
• Rank: $1$

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

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

Weierstrass equation

${y}^2={x}^{3}-{x}^{2}+\left(-13179a+36502\right){x}+2179225a+46814$

This is a global minimal model.

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-\frac{19827}{169} a + \frac{6819}{169} : \frac{2948}{2197} a - \frac{3246}{2197} : 1\right)$	$2.6573849845160811132831303366818796059$	$\infty$

Invariants

Conductor:	$\frak{N}$	=	$(256a-48)$	=	$(2)^{4}\cdot(-3a+1)\cdot(6a-5)$
Conductor norm:	$N(\frak{N})$	=	$55552$	=	$4^{4}\cdot7\cdot31$
Discriminant:	$\Delta$	=	$-46137344a-48234496$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-46137344a-48234496)$	=	$(2)^{21}\cdot(-3a+1)^{2}\cdot(6a-5)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$6680632650366976$	=	$4^{21}\cdot7^{2}\cdot31$
j-invariant:	$j$	=	$-\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728}$
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:	$\mathrm{Reg}(E/K)$	≈	$2.6573849845160811132831303366818796059$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$5.3147699690321622265662606733637592118$
Global period:	$\Omega(E/K)$	≈	$0.248523457545471556287579096485715738800$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $2\cdot2\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.0503608854677009156529063071878281616$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}3.050360885 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.248523 \cdot 5.314770 \cdot 4 } { {1^2 \cdot 1.732051} } \\ & \approx 3.050360885 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2)$	$4$	$2$	$I_{13}^{*}$	Additive	$1$	$4$	$21$	$9$
$(-3a+1)$	$7$	$2$	$I_{2}$	Non-split multiplicative	$1$	$1$	$2$	$2$
$(6a-5)$	$31$	$1$	$I_{1}$	Non-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
$3$	3B[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 55552.2-e consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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