Elliptic curve with Cremona label 137904bi1 (LMFDB label 137904.p2)

• Label: 137904bi1
• Conductor: $137904$
• Discriminant: $-458944512$
• j-invariant: $\frac{42875}{51}$
• CM: no
• Rank: $2$
• Torsion structure: $\Z/{2}\Z$

Minimal Weierstrass equation

$y^2=x^3-x^2+152x+688$

(, )

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(9, 52)$	$1.7073080470811687859589238509$	$\infty$
$(-3, 14)$	$2.6809145883638046169930132056$	$\infty$
$(-4, 0)$	$0$	$2$

Integral points

$\left(-4, 0\right)$, $(-3,\pm 14)$, $(9,\pm 52)$, $(12,\pm 64)$, $(204,\pm 2912)$

Invariants

Conductor:	$N$	=	$137904$	=	$2^{4} \cdot 3 \cdot 13^{2} \cdot 17$
Discriminant:	$\Delta$	=	$-458944512$	=	$-1 \cdot 2^{12} \cdot 3 \cdot 13^{3} \cdot 17$
j-invariant:	$j$	=	$\frac{42875}{51}$	=	$3^{-1} \cdot 5^{3} \cdot 7^{3} \cdot 17^{-1}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.34827177409797184788531393600$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.98611274582735764554529004585$
$abc$ quality:	$Q$	≈	$0.7681773811594625$
Szpiro ratio:	$\sigma_{m}$	≈	$2.2589091843783526$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$4.4278892677568428349479388616$
Real period:	$\Omega$	≈	$1.1135932646732980776799996150$
Tamagawa product:	$\prod_{p}c_p$	=	$8$  = $2^{2}\cdot1\cdot2\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(2)}(E,1)/2!$	≈	$9.8617353305864038061305437773$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 9.861735331 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.113593 \cdot 4.427889 \cdot 8}{2^2} \approx 9.861735331$

Modular invariants

Modular form 137904.2.a.p

$q - q^{3} - 4 q^{7} + q^{9} + 2 q^{11} + q^{17} - 2 q^{19} + O(q^{20})$

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

Modular degree:	49152
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{4}^{*}$	additive	-1	4	12	0
$3$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$13$	$2$	$III$	additive	-1	2	3	0
$17$	$1$	$I_{1}$	split 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	2.3.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $2652 = 2^{2} \cdot 3 \cdot 13 \cdot 17$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 665 & 1990 \\ 1988 & 663 \end{array}\right),\left(\begin{array}{rr} 1094 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 2649 & 4 \\ 2648 & 5 \end{array}\right),\left(\begin{array}{rr} 2452 & 1 \\ 2039 & 0 \end{array}\right),\left(\begin{array}{rr} 886 & 1 \\ 883 & 0 \end{array}\right)$.

The torsion field $K:=\Q(E[2652])$ is a degree-$788363476992$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2652\Z)$.

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
$2$	additive	$2$	$663 = 3 \cdot 13 \cdot 17$
$3$	nonsplit multiplicative	$4$	$45968 = 2^{4} \cdot 13^{2} \cdot 17$
$13$	additive	$50$	$816 = 2^{4} \cdot 3 \cdot 17$
$17$	split multiplicative	$18$	$8112 = 2^{4} \cdot 3 \cdot 13^{2}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 137904bi consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 8619m1, its twist by $-4$.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	$\Q(\sqrt{-663})$	$\Z/2\Z \oplus \Z/2\Z$	not in database
$4$	4.2.1792752.2	$\Z/4\Z$	not in database
$8$	8.0.8359509266843904.42	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	deg 8	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	deg 8	$\Z/6\Z$	not in database
$16$	deg 16	$\Z/8\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/6\Z$	not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	nonsplit	ss	ord	ord	add	split	ord	ss	ord	ord	ord	ss	ord	ord
$\lambda$-invariant(s)	-	2	2,2	2	2	-	3	2	2,2	2	2	2	2,2	4	2
$\mu$-invariant(s)	-	0	0,0	0	0	-	0	0	0,0	0	0	0	0,0	0	0

An entry - indicates that the invariants are not computed because the reduction is additive.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.