Elliptic curve 56.1-d1 over number field Q(14)\Q(\sqrt{14})

• Label: 2.2.56.1-56.1-d1
• Base field: $\Q(\sqrt{14})$
• Conductor norm: $56$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $2$

Base field $\Q(\sqrt{14})$

Generator $a$, with minimal polynomial $x^{2} - 14$; class number $1$.

Weierstrass equation

${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(150a-635\right){x}+2318a-9041$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{19598}{4225} a - \frac{6947}{1690} : -\frac{4630381}{1098500} a - \frac{29693197}{549250} : 1\right)$	$5.9221351593970926978791183875431340153$	$\infty$
$\left(\frac{156337}{8450} a + \frac{807133}{16900} : \frac{162030767}{1098500} a + \frac{1128405053}{2197000} : 1\right)$	$5.9221351593970926978791183875431340406$	$\infty$
$\left(4 a - \frac{13}{2} : \frac{11}{4} a - 28 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(2a)$	=	$(-a+4)^{3}\cdot(-2a+7)$
Conductor norm:	$N(\frak{N})$	=	$56$	=	$2^{3}\cdot7$
Discriminant:	$\Delta$	=	$32a$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(32a)$	=	$(-a+4)^{11}\cdot(-2a+7)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-14336$	=	$-2^{11}\cdot7$
j-invariant:	$j$	=	$-\frac{39775849362076815}{7} a + 21261085797246696$
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:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$32.363353551208464212080134907405718051$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$129.45341420483385684832053962962287220$
Global period:	$\Omega(E/K)$	≈	$0.65907338218172640606237416363638825668$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$  =  $1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.8503177441011686065450864444986300260$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 2.850317744 \approx L^{(2)}(E/K,1)/2! \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 0.659073 \cdot 129.453414 \cdot 1 } { {2^2 \cdot 7.483315} } \approx 2.850317744$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-a+4)$	$2$	$1$	$II^{*}$	Additive	$-1$	$3$	$11$	$0$
$(-2a+7)$	$7$	$1$	$I_{1}$	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
$2$	2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 56.1-d consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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