Elliptic curve 144.4-d3 over number field Q(17)\Q(\sqrt{17})

• Label: 2.2.17.1-144.4-d3
• Base field: $\Q(\sqrt{17})$
• Conductor norm: $144$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{17})$

Generator $a$, with minimal polynomial $x^{2} - x - 4$; class number $1$.

Weierstrass equation

${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-6a-8\right){x}+5a+8$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(0 : a + 2 : 1\right)$	$0.45883412381969792641265125870972462092$	$\infty$
$\left(\frac{3}{4} a + 1 : -\frac{7}{8} a - \frac{3}{2} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(3a+12)$	=	$(-a+2)^{4}\cdot(3)$
Conductor norm:	$N(\frak{N})$	=	$144$	=	$2^{4}\cdot9$
Discriminant:	$\Delta$	=	$1599a+396$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(1599a+396)$	=	$(-a+2)^{20}\cdot(3)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-9437184$	=	$-2^{20}\cdot9$
j-invariant:	$j$	=	$\frac{1095125}{768} a + \frac{2055079}{768}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.45883412381969792641265125870972462092$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.917668247639395852825302517419449241840$
Global period:	$\Omega(E/K)$	≈	$10.041230522034847068511366620133570423$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $2^{2}\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.2348489837483986559902860541349287202$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}2.234848984 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 10.041231 \cdot 0.917668 \cdot 4 } { {2^2 \cdot 4.123106} } \\ & \approx 2.234848984 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-a+2)$	$2$	$4$	$I_{12}^{*}$	Additive	$-1$	$4$	$20$	$8$
$(3)$	$9$	$1$	$I_{1}$	Non-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 144.4-d consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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