Elliptic curve 539.1-e1 over number field Q(33)\Q(\sqrt{33})

• Label: 2.2.33.1-539.1-e1
• Base field: $\Q(\sqrt{33})$
• Conductor norm: $539$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{33})$

Generator $a$, with minimal polynomial $x^{2} - x - 8$; class number $1$.

Weierstrass equation

${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1280a+3040\right){x}-161701a-383601$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{205}{9} a + \frac{479}{9} : -\frac{2417}{9} a - \frac{17158}{27} : 1\right)$	$1.1839287023003136282328748766856106169$	$\infty$
$\left(13 a + 30 : -7 a - 15 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-28a-63)$	=	$(-4a-9)\cdot(7)$
Conductor norm:	$N(\frak{N})$	=	$539$	=	$11\cdot49$
Discriminant:	$\Delta$	=	$-41503$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-41503)$	=	$(-4a-9)^{4}\cdot(7)^{3}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$1722499009$	=	$11^{4}\cdot49^{3}$
j-invariant:	$j$	=	$\frac{4657463}{41503}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.1839287023003136282328748766856106169$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$2.3678574046006272564657497533712212338$
Global period:	$\Omega(E/K)$	≈	$3.2101873408696119145423660433455397356$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$6$  =  $2\cdot3$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.9848158161738105597445225579762314471$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 1.984815816 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 3.210187 \cdot 2.367857 \cdot 6 } { {2^2 \cdot 5.744563} } \approx 1.984815816$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-4a-9)$	$11$	$2$	$I_{4}$	Non-split multiplicative	$1$	$1$	$4$	$4$
$(7)$	$49$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$

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 539.1-e consists of curves linked by isogenies of degree 2.

Base change

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

Base field	Curve
$\Q$	693.a2
$\Q$	847.a2