Elliptic curve 36.4-c3 over number field Q(−71)\Q(\sqrt{-71})

• Label: 2.0.71.1-36.4-c3
• Base field: $\Q(\sqrt{-71})$
• Conductor norm: $36$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $3$
• Rank: $1$

Base field $\Q(\sqrt{-71})$

Generator $a$, with minimal polynomial $x^{2} - x + 18$; class number $7$.

Weierstrass equation

${y}^2+{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(-16a-131\right){x}-149a-349$

This is not a global minimal model: it is minimal at all primes except $(5,a+1)$. No global minimal model exists.

Mordell-Weil group structure

$\Z \oplus \Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-\frac{29059}{42436} a - \frac{250583}{42436} : \frac{2124055}{8741816} a + \frac{26863499}{8741816} : 1\right)$	$5.2523051886693744665081747252624554907$	$\infty$
$\left(-a - 5 : -a - 2 : 1\right)$	$0$	$3$

Invariants

Conductor:	$\frak{N}$	=	$(18,2a)$	=	$(2,a)\cdot(2,a+1)\cdot(3,a)^{2}$
Conductor norm:	$N(\frak{N})$	=	$36$	=	$2\cdot2\cdot3^{2}$
Discriminant:	$\Delta$	=	$789184a+184824$
Discriminant ideal:	$(\Delta)$	=	$(789184a+184824)$	=	$(2,a)^{3}\cdot(2,a+1)^{3}\cdot(3,a)^{6}\cdot(5,a+1)^{12}$
Discriminant norm:	$N(\Delta)$	=	$11390625000000$	=	$2^{3}\cdot2^{3}\cdot3^{6}\cdot5^{12}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(5832,8a+3096)$	=	$(2,a)^{3}\cdot(2,a+1)^{3}\cdot(3,a)^{6}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$46656$	=	$2^{3}\cdot2^{3}\cdot3^{6}$
j-invariant:	$j$	=	$\frac{857375}{8}$
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:	$\mathrm{Reg}(E/K)$	≈	$5.2523051886693744665081747252624554907$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$10.504610377338748933016349450524910981$
Global period:	$\Omega(E/K)$	≈	$8.6259178240387304854320025847014532740$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$3$  =  $3\cdot1\cdot1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$3$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.5845515974140687728821915487463018989$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}3.584551597 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 8.625918 \cdot 10.504610 \cdot 3 } { {3^2 \cdot 8.426150} } \\ & \approx 3.584551597 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ 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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$
$(2,a+1)$	$2$	$1$	$I_{3}$	Non-split multiplicative	$1$	$1$	$3$	$3$
$(3,a)$	$3$	$1$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$
$(5,a+1)$	$5$	$1$	$I_0$	Good	$1$	$0$	$0$	$0$

Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p < 1000$ except those listed.

prime	Image of Galois Representation
$3$	3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 36.4-c consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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