Elliptic curve with Cremona label 1216b2 (LMFDB label 1216.e3)

• Label: 1216b2
• Conductor: $1216$
• Discriminant: $-920599396352$
• j-invariant: $\frac{94196375}{3511808}$
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2=x^3-x^2+607x+45601$

(, )

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(57, 512)$	$0.83477609491361802989130534462$	$\infty$

Integral points

$(-13,\pm 188)$, $(57,\pm 512)$

Invariants

Conductor:	$N$	=	$1216$	=	$2^{6} \cdot 19$
Discriminant:	$\Delta$	=	$-920599396352$	=	$-1 \cdot 2^{27} \cdot 19^{3}$
j-invariant:	$j$	=	$\frac{94196375}{3511808}$	=	$2^{-9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3} \cdot 19^{-3}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.97558423746310669938018748380$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.064136533376811264745660698387$
$abc$ quality:	$Q$	≈	$1.01874591611381$
Szpiro ratio:	$\sigma_{m}$	≈	$4.925493493116128$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.83477609491361802989130534462$
Real period:	$\Omega$	≈	$0.66843943526102164786503437766$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $2^{2}\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L'(E,1)$	≈	$2.2319890458138393663961927064$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\begin{aligned} 2.231989046 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.668439 \cdot 0.834776 \cdot 4}{1^2} \\ & \approx 2.231989046\end{aligned}$

Modular invariants

Modular form   1216.2.a.e

$q - q^{3} - q^{7} - 2 q^{9} + 6 q^{11} - 5 q^{13} + 3 q^{17} - q^{19} + O(q^{20})$

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

Modular degree:	1152
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{17}^{*}$	additive	1	6	27	9
$19$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3

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
$3$	3Cs	9.36.0.2

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $4104 = 2^{3} \cdot 3^{3} \cdot 19$, index $1296$, genus $43$, and generators

$\left(\begin{array}{rr} 2051 & 4050 \\ 2025 & 2645 \end{array}\right),\left(\begin{array}{rr} 19 & 54 \\ 792 & 3547 \end{array}\right),\left(\begin{array}{rr} 2377 & 54 \\ 2211 & 2755 \end{array}\right),\left(\begin{array}{rr} 2509 & 3512 \\ 912 & 2813 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3079 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 3918 & 1111 \end{array}\right)$.

The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.

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	$4$	$19$
$3$	good	$2$	$64 = 2^{6}$
$19$	nonsplit multiplicative	$20$	$64 = 2^{6}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 1216b consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

The minimal quadratic twist of this elliptic curve is 38a1, its twist by $8$.

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
$2$	$\Q(\sqrt{2})$	$\Z/3\Z$	2.2.8.1-722.1-b3
$2$	$\Q(\sqrt{-6})$	$\Z/3\Z$	not in database
$3$	3.1.152.1	$\Z/2\Z$	not in database
$4$	$\Q(\sqrt{2}, \sqrt{-3})$	$\Z/3\Z \oplus \Z/3\Z$	not in database
$6$	6.0.3511808.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$6$	6.2.184832.1	$\Z/6\Z$	not in database
$6$	6.0.4990464.1	$\Z/6\Z$	not in database
$12$	12.2.119973433931988992.9	$\Z/4\Z$	not in database
$12$	12.0.24904730935296.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$12$	12.0.12332795428864.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$12$	12.0.8990607867641856.2	$\Z/2\Z \oplus \Z/6\Z$	not in database
$18$	18.6.216561186743676164093509632.1	$\Z/9\Z$	not in database
$18$	18.0.2265306503399694103070297070602551296.3	$\Z/9\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	ord	ss	ord	ord	ord	ord	nonsplit	ord	ord	ord	ord	ss	ord	ss
$\lambda$-invariant(s)	-	1	1,1	1	1	1	1	3	1	1	1	1	1,1	1	1,1
$\mu$-invariant(s)	-	1	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.