Elliptic curve 81225.1-d1 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-81225.1-d1
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $81225$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-30a+77\right){x}-184a-9$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(a + 2 : -9 a + 15 : 1\right)$	$0.26414711843010366022554164086616837906$	$\infty$
$\left(\frac{19}{4} a - \frac{7}{4} : -\frac{3}{2} a + \frac{15}{8} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(315a-75)$	=	$(-2a+1)^{2}\cdot(-5a+3)^{2}\cdot(5)$
Conductor norm:	$N(\frak{N})$	=	$81225$	=	$3^{2}\cdot19^{2}\cdot25$
Discriminant:	$\Delta$	=	$147825a+34425$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(147825a+34425)$	=	$(-2a+1)^{8}\cdot(-5a+3)^{3}\cdot(5)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$28126186875$	=	$3^{8}\cdot19^{3}\cdot25^{2}$
j-invariant:	$j$	=	$\frac{15794551}{75} a - \frac{61746}{5}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.26414711843010366022554164086616837906$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.528294236860207320451083281732336758120$
Global period:	$\Omega(E/K)$	≈	$2.6960186431141276564443822161800005310$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$16$  =  $2^{2}\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.2892594268040393663570959445686674939$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 3.289259427 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 2.696019 \cdot 0.528294 \cdot 16 } { {2^2 \cdot 1.732051} } \approx 3.289259427$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$4$	$I_{2}^{*}$	Additive	$-1$	$2$	$8$	$2$
$(-5a+3)$	$19$	$2$	$III$	Additive	$1$	$2$	$3$	$0$
$(5)$	$25$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$

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 81225.1-d consists of curves linked by isogenies of degree 2.

Base change

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

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