Elliptic curve 16.1-b1 over number field Q(26)\Q(\sqrt{26})

• Label: 2.2.104.1-16.1-b1
• Base field: $\Q(\sqrt{26})$
• Conductor norm: $16$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $1$
• Rank: $0$

Base field $\Q(\sqrt{26})$

Generator $a$, with minimal polynomial $x^{2} - 26$; class number $2$.

Weierstrass equation

${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(9586a-48883\right){x}-1173397a+5983175$

This is not a global minimal model: it is minimal at all primes except $(2,a)$. No global minimal model exists.

Mordell-Weil group structure

trivial

Invariants

Conductor:	$\frak{N}$	=	$(4)$	=	$(2,a)^{4}$
Conductor norm:	$N(\frak{N})$	=	$16$	=	$2^{4}$
Discriminant:	$\Delta$	=	$-4096$
Discriminant ideal:	$(\Delta)$	=	$(-4096)$	=	$(2,a)^{24}$
Discriminant norm:	$N(\Delta)$	=	$16777216$	=	$2^{24}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(64)$	=	$(2,a)^{12}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$4096$	=	$2^{12}$
j-invariant:	$j$	=	$-23788477376$
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:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$21.289112115898858603301341815485675910$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.0875691943467857742256996067683401680$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 2.087569194 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 21.289112 \cdot 1 \cdot 1 } { {1^2 \cdot 10.198039} } \approx 2.087569194$

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $\frak{p}$ 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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$1$	$II^{*}$	Additive	$1$	$4$	$12$	$0$

Galois Representations

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

prime	Image of Galois Representation
$3$	3Nn
$5$	5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 16.1-b 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.