Elliptic curve 128.7-d2 over number field Q(−79)\Q(\sqrt{-79})

• Label: 2.0.79.1-128.7-d2
• Base field: $\Q(\sqrt{-79})$
• Conductor norm: $128$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $1$
• Rank: $1$

Base field $\Q(\sqrt{-79})$

Generator $a$, with minimal polynomial $x^{2} - x + 20$; class number $5$.

Weierstrass equation

${y}^2+\left(a+1\right){x}{y}={x}^3-a{x}^2+\left(30a-98\right){x}+152a+712$

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

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-28 : 48 a - 16 : 1\right)$	$1.4377781612433562159879100404601488830$	$\infty$

Invariants

Conductor:	$\frak{N}$	=	$(2a+6)$	=	$(2,a)\cdot(2,a+1)^{6}$
Conductor norm:	$N(\frak{N})$	=	$128$	=	$2\cdot2^{6}$
Discriminant:	$\Delta$	=	$-113708444a-25512020$
Discriminant ideal:	$(\Delta)$	=	$(-113708444a-25512020)$	=	$(2,a)^{2}\cdot(2,a+1)^{28}\cdot(5,a)^{12}$
Discriminant norm:	$N(\Delta)$	=	$262144000000000000$	=	$2^{2}\cdot2^{28}\cdot5^{12}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(268435456,4a+159822988)$	=	$(2,a)^{2}\cdot(2,a+1)^{28}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$1073741824$	=	$2^{2}\cdot2^{28}$
j-invariant:	$j$	=	$-\frac{179945}{1024} a + \frac{111409}{256}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.4377781612433562159879100404601488830$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$2.8755563224867124319758200809202977660$
Global period:	$\Omega(E/K)$	≈	$4.8581560426924287306412954705103335460$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $2\cdot2\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$6.2869467827912017212775958016689696145$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 6.286946783 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 4.858156 \cdot 2.875556 \cdot 4 } { {1^2 \cdot 8.888194} } \approx 6.286946783$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$
$(2,a+1)$	$2$	$2$	$I_{18}^{*}$	Additive	$-1$	$6$	$28$	$10$
$(5,a)$	$5$	$1$	$I_0$	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
$5$	5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 128.7-d consists of curves linked by isogenies of degree 5.

Base change

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

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