Elliptic curve 28512.10-e2 over number field Q(−7)\Q(\sqrt{-7})

• Label: 2.0.7.1-28512.10-e2
• Base field: $\Q(\sqrt{-7})$
• Conductor norm: $28512$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $4$
• Rank: $1$

Base field $\Q(\sqrt{-7})$

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

Weierstrass equation

${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-93a+95\right){x}+31a-549$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{4}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(a + 5 : 6 a - 22 : 1\right)$	$1.1255308795313739570025835875943139293$	$\infty$
$\left(a - 3 : -18 a + 2 : 1\right)$	$0$	$4$

Invariants

Conductor:	$\frak{N}$	=	$(54a+126)$	=	$(a)\cdot(-a+1)^{4}\cdot(3)^{2}\cdot(2a+1)$
Conductor norm:	$N(\frak{N})$	=	$28512$	=	$2\cdot2^{4}\cdot9^{2}\cdot11$
Discriminant:	$\Delta$	=	$-17915904a-8957952$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-17915904a-8957952)$	=	$(a)^{12}\cdot(-a+1)^{12}\cdot(3)^{7}\cdot(2a+1)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$882693944377344$	=	$2^{12}\cdot2^{12}\cdot9^{7}\cdot11$
j-invariant:	$j$	=	$\frac{837849187}{135168} a - \frac{663012145}{135168}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.1255308795313739570025835875943139293$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$2.2510617590627479140051671751886278586$
Global period:	$\Omega(E/K)$	≈	$1.42930189168260703530188978819338700178$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$96$  =  $( 2^{2} \cdot 3 )\cdot2\cdot2^{2}\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$7.2964835744024663042183504001996404976$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}7.296483574 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.429302 \cdot 2.251062 \cdot 96 } { {4^2 \cdot 2.645751} } \\ & \approx 7.296483574 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$12$	$I_{12}$	Split multiplicative	$-1$	$1$	$12$	$12$
$(-a+1)$	$2$	$2$	$I_{4}^{*}$	Additive	$-1$	$4$	$12$	$0$
$(3)$	$9$	$4$	$I_{1}^{*}$	Additive	$1$	$2$	$7$	$1$
$(2a+1)$	$11$	$1$	$I_{1}$	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 and 4.
Its isogeny class 28512.10-e consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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