Elliptic curve with Cremona label 42135j1 (LMFDB label 42135.b1)

• Label: 42135j1
• Conductor: $42135$
• Discriminant: $266635546875$
• j-invariant: $\frac{185025936889}{94921875}$
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy=x^3-1675x+8750$

(, )

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-25, 200)$	$0.10696439187867924280674537763$	$\infty$

Integral points

$\left(-35, 175\right)$, $\left(-35, -140\right)$, $\left(-25, 200\right)$, $\left(-25, -175\right)$, $\left(5, 20\right)$, $\left(5, -25\right)$, $\left(50, 200\right)$, $\left(50, -250\right)$, $\left(1250, 43550\right)$, $\left(1250, -44800\right)$

Invariants

Conductor:	$N$	=	$42135$	=	$3 \cdot 5 \cdot 53^{2}$
Discriminant:	$\Delta$	=	$266635546875$	=	$3^{5} \cdot 5^{8} \cdot 53^{2}$
j-invariant:	$j$	=	$\frac{185025936889}{94921875}$	=	$3^{-5} \cdot 5^{-8} \cdot 37^{3} \cdot 41^{3} \cdot 53$
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:	$h_{\mathrm{Faltings}}$	≈	$0.88465140571398430553408337962$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.22293608678863066651000518978$
$abc$ quality:	$Q$	≈	$0.9760671044978608$
Szpiro ratio:	$\sigma_{m}$	≈	$3.182036831104157$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.10696439187867924280674537763$
Real period:	$\Omega$	≈	$0.86460408072072114515826773261$
Tamagawa product:	$\prod_{p}c_p$	=	$40$  = $5\cdot2^{3}\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L'(E,1)$	≈	$3.6992739884046574939192848702$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 3.699273988 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.864604 \cdot 0.106964 \cdot 40}{1^2} \approx 3.699273988$

Modular invariants

Modular form   42135.2.a.b

$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - 5 q^{13} + 2 q^{14} + q^{15} - q^{16} + 4 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$

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

Modular degree:	43200
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$3$	$5$	$I_{5}$	split multiplicative	-1	1	5	5
$5$	$8$	$I_{8}$	split multiplicative	-1	1	8	8
$53$	$1$	$II$	additive	1	2	2	0

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 12.2.0.a.1, level $12 = 2^{2} \cdot 3$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 11 & 0 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[12])$ is a degree-$2304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12\Z)$.

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
$2$	good	$2$	$8427 = 3 \cdot 53^{2}$
$3$	split multiplicative	$4$	$14045 = 5 \cdot 53^{2}$
$5$	split multiplicative	$6$	$2809 = 53^{2}$
$53$	additive	$522$	$15 = 3 \cdot 5$

Isogenies

This curve has no rational isogenies. Its isogeny class 42135j consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.3.33708.1	$\Z/2\Z$	not in database
$6$	6.6.13634751168.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	deg 8	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	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	53
Reduction type	ord	split	split	ord	ord	ord	ord	ord	ord	ord	ord	ord	ss	ord	ord	add
$\lambda$-invariant(s)	14	2	2	1	1	1	1	1	1	1	1	1	1,1	1	1	-
$\mu$-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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.