Elliptic curve 135.1-g4 over number field Q(10)\Q(\sqrt{10})

• Label: 2.2.40.1-135.1-g4
• Base field: $\Q(\sqrt{10})$
• Conductor norm: $135$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{10})$

Generator $a$, with minimal polynomial $x^{2} - 10$; class number $2$.

Weierstrass equation

${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(68a-266\right){x}+506a-1784$

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

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{11}{9} a - \frac{79}{9} : -\frac{13}{27} a + \frac{50}{27} : 1\right)$	$2.4917886595952619109030459294972334088$	$\infty$
$\left(-5 a + 11 : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-3a-15)$	=	$(3,a+1)\cdot(3,a+2)^{2}\cdot(5,a)$
Conductor norm:	$N(\frak{N})$	=	$135$	=	$3\cdot3^{2}\cdot5$
Discriminant:	$\Delta$	=	$696000a+1032000$
Discriminant ideal:	$(\Delta)$	=	$(696000a+1032000)$	=	$(2,a)^{12}\cdot(3,a+1)\cdot(3,a+2)^{9}\cdot(5,a)^{6}$
Discriminant norm:	$N(\Delta)$	=	$3779136000000$	=	$2^{12}\cdot3\cdot3^{9}\cdot5^{6}$
Minimal discriminant:	$\frak{D}_{\mathrm{min}}$	=	$(10875a+16125)$	=	$(3,a+1)\cdot(3,a+2)^{9}\cdot(5,a)^{6}$
Minimal discriminant norm:	$N(\frak{D}_{\mathrm{min}})$	=	$922640625$	=	$3\cdot3^{9}\cdot5^{6}$
j-invariant:	$j$	=	$\frac{66006784}{375} a + \frac{226222784}{375}$
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:	$\mathrm{Reg}(E/K)$	≈	$2.4917886595952619109030459294972334088$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$4.9835773191905238218060918589944668176$
Global period:	$\Omega(E/K)$	≈	$3.6098533029722432540925705659458719723$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $1\cdot1\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.8444660747057187919392215700267920443$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}2.844466075 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.609853 \cdot 4.983577 \cdot 4 } { {2^2 \cdot 6.324555} } \\ & \approx 2.844466075 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$1$	$I_0$	Good	$1$	$0$	$0$	$0$
$(3,a+1)$	$3$	$1$	$I_{1}$	Split multiplicative	$-1$	$1$	$1$	$1$
$(3,a+2)$	$3$	$2$	$III^{*}$	Additive	$1$	$2$	$9$	$0$
$(5,a)$	$5$	$2$	$I_{6}$	Non-split multiplicative	$1$	$1$	$6$	$6$

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
$3$	3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 135.1-g consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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