Elliptic curve with Cremona label 10780e1 (LMFDB label 10780.e2)

• Label: 10780e1
• Conductor: $10780$
• Discriminant: $-189728000$
• j-invariant: \( \frac{16674224}{15125} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3-x^2+124x-440\)  

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\(\left(6, 22\right)\)
\(\hat{h}(P)\)	≈	$0.52852977215470043589847081195$

Integral points

\((6,\pm 22)\), \((14,\pm 62)\)  

Invariants

Conductor:	\( 10780 \)	=	\(2^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Discriminant:	\(-189728000 \)	=	\(-1 \cdot 2^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11^{2} \)
j-invariant:	\( \frac{16674224}{15125} \)	=	\(2^{4} \cdot 5^{-3} \cdot 7 \cdot 11^{-2} \cdot 53^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(0.27576205734751833231739937462\dots\)
Stable Faltings height:	\(-0.51065442120166409147831416359\dots\)

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(0.52852977215470043589847081195\dots\)
Real period:	\(0.98380206584854091661480009223\dots\)
Tamagawa product:	\( 6 \)  = \( 3\cdot1\cdot1\cdot2 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 10780.2.a.e

\( q - q^{3} - q^{5} - 2q^{9} - q^{11} + 4q^{13} + q^{15} - 6q^{17} - 2q^{19} + O(q^{20}) \)  

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

Modular degree:	3456
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 3.1198120902495175536940600163151519802 \)

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\((j)_{-}\)
\(2\)	\(3\)	\(IV^{*}\)	Additive	-1	2	8	0
\(5\)	\(1\)	\(I_{3}\)	Non-split multiplicative	1	1	3	3
\(7\)	\(1\)	\(II\)	Additive	-1	2	2	0
\(11\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2

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 \( p \) except those listed.

prime	Image of Galois representation
\(3\)	B

$p$-adic data

$p$-adic regulators

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

Iwasawa invariants

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

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

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 10780e consists of 2 curves linked by isogenies of degree 3.

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
$2$	\(\Q(\sqrt{-7}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.980.1	\(\Z/2\Z\)	Not in database
$6$	6.0.19208000.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.2.106302795984.2	\(\Z/3\Z\)	Not in database
$6$	6.0.6722800.1	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	Deg 12	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$18$	18.0.741890158134787801407807000000000000.1	\(\Z/9\Z\)	Not in database
$18$	18.2.4805007323001463790159723917295616000000.1	\(\Z/6\Z\)	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.