Elliptic curve 3042.1-e3 over number field Q(3)\Q(\sqrt{3})

• Label: 2.2.12.1-3042.1-e3
• Base field: $\Q(\sqrt{3})$
• Conductor norm: $3042$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $4$
• Rank: $0$

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

Generator $a$, with minimal polynomial $x^{2} - 3$; class number $1$.

Weierstrass equation

${y}^2+a{x}{y}={x}^{3}-63{x}-209$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-4 a + 2 : -a + 6 : 1\right)$	$0$	$2$
$\left(-\frac{19}{4} : \frac{19}{8} a : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(39a+39)$	=	$(a+1)\cdot(a)^{2}\cdot(a+4)\cdot(a-4)$
Conductor norm:	$N(\frak{N})$	=	$3042$	=	$2\cdot3^{2}\cdot13\cdot13$
Discriminant:	$\Delta$	=	$18252$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(18252)$	=	$(a+1)^{4}\cdot(a)^{6}\cdot(a+4)^{2}\cdot(a-4)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$333135504$	=	$2^{4}\cdot3^{6}\cdot13^{2}\cdot13^{2}$
j-invariant:	$j$	=	$\frac{1033364331}{676}$
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:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$2.8675937787356667391714538161717241907$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$64$  =  $2^{2}\cdot2^{2}\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.3112120801590666686988426655417238167$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 3.311212080 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 2.867594 \cdot 1 \cdot 64 } { {4^2 \cdot 3.464102} } \approx 3.311212080$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a+1)$	$2$	$4$	$I_{4}$	Split multiplicative	$-1$	$1$	$4$	$4$
$(a)$	$3$	$4$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$
$(a+4)$	$13$	$2$	$I_{2}$	Non-split multiplicative	$1$	$1$	$2$	$2$
$(a-4)$	$13$	$2$	$I_{2}$	Non-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
$2$	2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 3042.1-e consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a $\Q$-curve. It is the base change of the following 2 elliptic curves:

Base field	Curve
$\Q$	234.d1
$\Q$	1872.g1