Elliptic curve with Cremona label 141120cg1 (LMFDB label 141120.po1)

• Label: 141120cg1
• Conductor: $141120$
• Discriminant: $-1.181\times 10^{17}$
• j-invariant: \( \frac{14155776}{84035} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3+56448x-15706656\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(\frac{10228225}{5329}, \frac{32925083471}{389017}\right) \)
\(\hat{h}(P)\)	≈	$11.841960095423768774396174636$

Integral points

None

Invariants

Conductor:	\( 141120 \)	=	\(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Discriminant:	\(-118085320347402240 \)	=	\(-1 \cdot 2^{14} \cdot 3^{6} \cdot 5 \cdot 7^{11} \)
j-invariant:	\( \frac{14155776}{84035} \)	=	\(2^{19} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-5}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(11.841960095423768774396174636\)
Real period:	\(0.16606788519823903698577599686\)
Tamagawa product:	\( 4 \)  = \( 1\cdot1\cdot1\cdot2^{2} \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 141120.2.a.po

\( q + q^{5} + 5q^{11} - 3q^{13} - q^{17} - 6q^{19} + O(q^{20}) \)

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

Modular degree:	1290240
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 7.8662770785958488977352433002301107449 \)

Local data

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

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(1\)	\(II^{*}\)	Additive	-1	6	14	0
\(3\)	\(1\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(5\)	\(1\)	\(I_{1}\)	Split multiplicative	-1	1	1	1
\(7\)	\(4\)	\(I_5^{*}\)	Additive	-1	2	11	5

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .

$p$-adic data

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	add	split	add	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	-	2	-	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

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

Isogenies

This curve has no rational isogenies. Its isogeny class 141120cg consists of this curve only.

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
$3$	3.1.140.1	\(\Z/2\Z\)	Not in database
$6$	6.0.686000.1	\(\Z/2\Z \times \Z/2\Z\)	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.