Elliptic curve with Cremona label 128778m2 (LMFDB label 128778.k1)

• Label: 128778m2
• Conductor: $128778$
• Discriminant: $1.852\times 10^{23}$
• j-invariant: \( \frac{1236526859255318155975783969}{38367061931916216} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+xy+y=x^3-3779026918x-89416900063840\)

(, )

Mordell-Weil group structure

\(\Z\)

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
\( \left(-\frac{10257816}{289}, \frac{93945008}{4913}\right) \)	$10.014422189854268540241101955$	$\infty$

Integral points

None

Invariants

Conductor:	$N$	=	\( 128778 \)	=	$2 \cdot 3 \cdot 13^{2} \cdot 127$
Minimal Discriminant:	$\Delta$	=	$185190479836530578634744$	=	$2^{3} \cdot 3^{2} \cdot 13^{6} \cdot 127^{7} $
j-invariant:	$j$	=	\( \frac{1236526859255318155975783969}{38367061931916216} \)	=	$2^{-3} \cdot 3^{-2} \cdot 73^{3} \cdot 127^{-7} \cdot 503^{3} \cdot 29231^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$3.9689484974721449987398769351$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$2.6864738187413766307131332143$
$abc$ quality:	$Q$	≈	$1.0641590672157408$
Szpiro ratio:	$\sigma_{m}$	≈	$6.609962854060507$
Intrinsic torsion order:	$\#E(\mathbb Q)_\text{tors}^\text{is}$	=	$$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$ 1$
Mordell-Weil rank:	$r$	=	$ 1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$10.014422189854268540241101955$
Real period:	$\Omega$	≈	$0.019255489309786093588769719061$
Tamagawa product:	$\prod_{p}c_p$	=	$ 28 $  = $ 1\cdot2\cdot2\cdot7 $
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$ L'(E,1)$	≈	$5.3993127837718582582525463217 $
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$$\begin{aligned} 5.399312784 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.019255 \cdot 10.014422 \cdot 28}{1^2} \\ & \approx 5.399312784\end{aligned}$$

Modular invariants

Modular form 128778.2.a.k

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

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

Modular degree:	60850944
$ \Gamma_0(N) $-optimal:	no
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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3
$3$	$2$	$I_{2}$	split multiplicative	-1	1	2	2
$13$	$2$	$I_0^{*}$	additive	1	2	6	0
$127$	$7$	$I_{7}$	split multiplicative	-1	1	7	7

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	$\ell$-adic index
$7$	7B.6.3	7.24.0.2	$24$

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 92456 = 2^{3} \cdot 7 \cdot 13 \cdot 127 \), index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 85343 & 0 \\ 0 & 92455 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 55121 & 9152 \\ 53326 & 22569 \end{array}\right),\left(\begin{array}{rr} 92443 & 14 \\ 92442 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 81537 & 85358 \\ 58695 & 42771 \end{array}\right),\left(\begin{array}{rr} 69343 & 85358 \\ 65793 & 42771 \end{array}\right),\left(\begin{array}{rr} 46229 & 85358 \\ 88907 & 42771 \end{array}\right)$.

The torsion field $K:=\Q(E[92456])$ is a degree-$218172079617343488$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/92456\Z)$.

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	$4$	\( 21463 = 13^{2} \cdot 127 \)
$3$	split multiplicative	$4$	\( 21463 = 13^{2} \cdot 127 \)
$7$	good	$2$	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
$13$	additive	$86$	\( 762 = 2 \cdot 3 \cdot 127 \)
$127$	split multiplicative	$128$	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 128778m consists of 2 curves linked by isogenies of degree 7.

Twists

The minimal quadratic twist of this elliptic curve is 762g2, its twist by $13$.

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
$3$	3.3.1016.1	\(\Z/2\Z\)	not in database
$6$	6.6.1048772096.1	\(\Z/2\Z \oplus \Z/2\Z\)	not in database
$6$	6.0.36924979.1	\(\Z/7\Z\)	not in database
$8$	deg 8	\(\Z/3\Z\)	not in database
$12$	deg 12	\(\Z/4\Z\)	not in database
$14$	14.2.92638396885886175157077823488.1	\(\Z/7\Z\)	not in database
$18$	18.0.55376183192601445167355674698721570586624.1	\(\Z/14\Z\)	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	127
Reduction type	nonsplit	split	ord	ord	ord	add	ord	ord	ord	ord	ord	ord	ord	ord	ord	split
$\lambda$-invariant(s)	7	2	1	1	1	-	1	1	1	1	1	1	1	1	1	2
$\mu$-invariant(s)	0	0	0	1	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.