Elliptic curve 16.1-f1 over number field Q(39)\Q(\sqrt{39})

• Label: 2.2.156.1-16.1-f1
• Base field: $\Q(\sqrt{39})$
• Conductor norm: $16$
• CM: yes ($-3$)
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $0$

Base field $\Q(\sqrt{39})$

Generator $a$, with minimal polynomial $x^{2} - 39$; class number $2$.

Weierstrass equation

${y}^2={x}^{3}-a{x}^{2}+13{x}-678917764269a+4239840078928$

This is not a global minimal model: it is minimal at all primes except $(2,a+1)$. No global minimal model exists.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(1633 a - 10196 : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(4)$	=	$(2,a+1)^{4}$
Conductor norm:	$N(\frak{N})$	=	$16$	=	$2^{4}$
Discriminant:	$\Delta$	=	$1024$
Discriminant ideal:	$(\Delta)$	=	$(1024)$	=	$(2,a+1)^{20}$
Discriminant norm:	$N(\Delta)$	=	$1048576$	=	$2^{20}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(16)$	=	$(2,a+1)^{8}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$256$	=	$2^{8}$
j-invariant:	$j$	=	$0$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-3})/2]$    (potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$

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:	$\Omega(E/K)$	≈	$17.695031908454309764234228747255048751$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.8334727910228962235088803246755598751$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$4$ (rounded)

BSD formula

$\displaystyle 2.833472791 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 4 \cdot 17.695032 \cdot 1 \cdot 2 } { {2^2 \cdot 12.489996} } \approx 2.833472791$

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a+1)$	$2$	$2$	$I_0^{*}$	Additive	$1$	$4$	$8$	$0$

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 16.1-f consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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