Elliptic curve with Cremona label 438702x1 (LMFDB label 438702.x4)

• Label: 438702x1
• Conductor: $438702$
• Discriminant: $-1.201\times 10^{15}$
• j-invariant: $-\frac{37159393753}{49741824}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{2}\Z$

Minimal Weierstrass equation

$y^2+xy+y=x^3-20092x-1996870$

(, )

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(177, -89)$	$0$	$2$

Integral points

$\left(177, -89\right)$

Invariants

Conductor:	$N$	=	$438702$	=	$2 \cdot 3 \cdot 11 \cdot 17^{2} \cdot 23$
Discriminant:	$\Delta$	=	$-1200646708985856$	=	$-1 \cdot 2^{16} \cdot 3 \cdot 11 \cdot 17^{6} \cdot 23$
j-invariant:	$j$	=	$-\frac{37159393753}{49741824}$	=	$-1 \cdot 2^{-16} \cdot 3^{-1} \cdot 11^{-1} \cdot 23^{-1} \cdot 47^{3} \cdot 71^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$1.5862039187206638394207988031$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.16959724669255579929603149416$
$abc$ quality:	$Q$	≈	$1.0219745635225241$
Szpiro ratio:	$\sigma_{m}$	≈	$3.2740948327187227$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.19104881082150717613651126270$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $2\cdot1\cdot1\cdot2\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$2$
Special value:	$L(E,1)$	≈	$0.76419524328602870454604505081$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$4$ = $2^2$    (exact)

BSD formula

$\displaystyle 0.764195243 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.191049 \cdot 1.000000 \cdot 4}{2^2} \approx 0.764195243$

Modular invariants

Modular form 438702.2.a.x

$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + 2 q^{10} - q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})$

For more coefficients, see the Downloads section to the right.

Modular degree:	1769472
$\Gamma_0(N)$-optimal:	not computed^* (one of 3 curves in this isogeny class which might be optimal)
Manin constant:	1 (conditional^*)

^* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that this curve is optimal.

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$I_{16}$	nonsplit multiplicative	1	1	16	16
$3$	$1$	$I_{1}$	split multiplicative	-1	1	1	1
$11$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$17$	$2$	$I_0^{*}$	additive	1	2	6	0
$23$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	4.6.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $103224 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 23$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 52276 & 66793 \\ 91103 & 24294 \end{array}\right),\left(\begin{array}{rr} 103217 & 8 \\ 103216 & 9 \end{array}\right),\left(\begin{array}{rr} 32641 & 32640 \\ 29614 & 20503 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 31468 & 66793 \\ 100487 & 24294 \end{array}\right),\left(\begin{array}{rr} 30359 & 0 \\ 0 & 103223 \end{array}\right),\left(\begin{array}{rr} 6835 & 6834 \\ 81226 & 72115 \end{array}\right),\left(\begin{array}{rr} 20248 & 97155 \\ 93109 & 66794 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 103218 & 103219 \end{array}\right)$.

The torsion field $K:=\Q(E[103224])$ is a degree-$424337074657689600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/103224\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	nonsplit multiplicative	$4$	$219351 = 3 \cdot 11 \cdot 17^{2} \cdot 23$
$3$	split multiplicative	$4$	$146234 = 2 \cdot 11 \cdot 17^{2} \cdot 23$
$11$	nonsplit multiplicative	$12$	$39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23$
$17$	additive	$146$	$1518 = 2 \cdot 3 \cdot 11 \cdot 23$
$23$	nonsplit multiplicative	$24$	$19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 438702x consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 1518f1, its twist by $17$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.