Elliptic curve 225.1-b4 over number field Q(57)\Q(\sqrt{57})

• Label: 2.2.57.1-225.1-b4
• Base field: $\Q(\sqrt{57})$
• Conductor norm: $225$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $4$
• Rank: $1$

Base field $\Q(\sqrt{57})$

Generator $a$, with minimal polynomial $x^{2} - x - 14$; class number $1$.

Weierstrass equation

${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(1203a-5136\right){x}+19185a-82011$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-58 a + \frac{987}{4} : \frac{4813}{4} a - \frac{41145}{8} : 1\right)$	$4.0277074581286009456359540907554223438$	$\infty$
$\left(-14 a + 58 : 7 a - 29 : 1\right)$	$0$	$2$
$\left(2 a - 10 : -a + 5 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(15)$	=	$(4a+13)^{2}\cdot(5)$
Conductor norm:	$N(\frak{N})$	=	$225$	=	$3^{2}\cdot25$
Discriminant:	$\Delta$	=	$1366875$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(1366875)$	=	$(4a+13)^{14}\cdot(5)^{4}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$1868347265625$	=	$3^{14}\cdot25^{4}$
j-invariant:	$j$	=	$\frac{111284641}{50625}$
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:	$\mathrm{Reg}(E/K)$	≈	$4.0277074581286009456359540907554223438$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$8.0554149162572018912719081815108446876$
Global period:	$\Omega(E/K)$	≈	$5.1631319424297510699594780670656647538$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$8$  =  $2^{2}\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.7544425258751457899834582203215905609$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 2.754442526 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 5.163132 \cdot 8.055415 \cdot 8 } { {4^2 \cdot 7.549834} } \approx 2.754442526$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(4a+13)$	$3$	$4$	$I_{8}^{*}$	Additive	$-1$	$2$	$14$	$8$
$(5)$	$25$	$2$	$I_{4}$	Non-split multiplicative	$1$	$1$	$4$	$4$

Galois Representations

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

prime	Image of Galois Representation
$2$	2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 225.1-b consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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