Elliptic curve with Cremona label 210e2 (LMFDB label 210.e6)

• Label: 210e2
• Conductor: $210$
• Discriminant: $51438240000$
• j-invariant: $\frac{135487869158881}{51438240000}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{2}\Z \oplus \Z/{8}\Z$

This is the minimal-conductor curve with torsion $(\Z/2\Z) \times (\Z/8\Z)$.

Minimal Weierstrass equation

$y^2+xy=x^3-1070x+7812$

(, )

Mordell-Weil group structure

$\Z/{2}\Z \oplus \Z/{8}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-36, 18)$	$0$	$2$
$(4, 58)$	$0$	$8$

Integral points

$\left(-36, 18\right)$, $\left(-26, 148\right)$, $\left(-26, -122\right)$, $\left(-8, 130\right)$, $\left(-8, -122\right)$, $\left(4, 58\right)$, $\left(4, -62\right)$, $\left(28, -14\right)$, $\left(34, 88\right)$, $\left(34, -122\right)$, $\left(64, 418\right)$, $\left(64, -482\right)$, $\left(244, 3658\right)$, $\left(244, -3902\right)$

Invariants

Conductor:	$N$	=	$210$	=	$2 \cdot 3 \cdot 5 \cdot 7$
Discriminant:	$\Delta$	=	$51438240000$	=	$2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2}$
j-invariant:	$j$	=	$\frac{135487869158881}{51438240000}$	=	$2^{-8} \cdot 3^{-8} \cdot 5^{-4} \cdot 7^{-2} \cdot 51361^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.75470864605564508282719006522$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.75470864605564508282719006522$
$abc$ quality:	$Q$	≈	$1.0190986885579905$
Szpiro ratio:	$\sigma_{m}$	≈	$6.085515026358872$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.0259330100195332631677131886$
Tamagawa product:	$\prod_{p}c_p$	=	$512$  = $2^{3}\cdot2^{3}\cdot2^{2}\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$16$
Special value:	$L(E,1)$	≈	$2.0518660200390665263354263771$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 2.051866020 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.025933 \cdot 1.000000 \cdot 512}{16^2} \approx 2.051866020$

Modular invariants

Modular form   210.2.a.e

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

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

Modular degree:	256
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$8$	$I_{8}$	split multiplicative	-1	1	8	8
$3$	$8$	$I_{8}$	split multiplicative	-1	1	8	8
$5$	$4$	$I_{4}$	split multiplicative	-1	1	4	4
$7$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2

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
$2$	2Cs	8.96.0.40

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 15 & 16 \\ 74 & 289 \end{array}\right),\left(\begin{array}{rr} 1121 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1353 & 16 \\ 1076 & 121 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 320 & 617 \end{array}\right),\left(\begin{array}{rr} 241 & 16 \\ 1446 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right)$.

The torsion field $K:=\Q(E[1680])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.

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
$2$	split multiplicative	$4$	$1$
$3$	split multiplicative	$4$	$70 = 2 \cdot 5 \cdot 7$
$5$	split multiplicative	$6$	$42 = 2 \cdot 3 \cdot 7$
$7$	nonsplit multiplicative	$8$	$30 = 2 \cdot 3 \cdot 5$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 210e consists of 8 curves linked by isogenies of degrees dividing 16.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{8}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$4$	$\Q(i, \sqrt{7})$	$\Z/4\Z \oplus \Z/8\Z$	not in database
$4$	$\Q(\sqrt{-6}, \sqrt{10})$	$\Z/2\Z \oplus \Z/16\Z$	not in database
$4$	$\Q(\sqrt{6}, \sqrt{70})$	$\Z/2\Z \oplus \Z/16\Z$	not in database
$8$	8.2.4253299470000.8	$\Z/2\Z \oplus \Z/24\Z$	not in database
$16$	deg 16	$\Z/8\Z \oplus \Z/8\Z$	not in database
$16$	16.0.5951500145509072896.2	$\Z/4\Z \oplus \Z/16\Z$	not in database
$16$	16.0.63456228123711897600000000.20	$\Z/4\Z \oplus \Z/16\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/32\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/32\Z$	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
Reduction type	split	split	split	nonsplit
$\lambda$-invariant(s)	2	5	1	0
$\mu$-invariant(s)	0	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

$p$-adic regulators

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

Additional information

This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/8\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $210 = 2 \cdot 3 \cdot 5 \cdot 7$ (for instance, if $E$ had good reduction at $7$ then the reduction mod $7$ would have at least $16$ points, which exceeds the Weil bound $(\sqrt7+1)^2 < 14$.