Elliptic curve with Cremona label 490k1 (LMFDB label 490.e1)

• Label: 490k1
• Conductor: $490$
• Discriminant: $-980$
• j-invariant: \( -\frac{5154200289}{20} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+xy+y=x^3-x^2-132x-549\)  

Mordell-Weil group structure

trivial

Integral points

\(\)  

Invariants

Conductor:	\( 490 \)	=	\(2 \cdot 5 \cdot 7^{2}\)
Discriminant:	\(-980 \)	=	\(-1 \cdot 2^{2} \cdot 5 \cdot 7^{2} \)
j-invariant:	\( -\frac{5154200289}{20} \)	=	\(-1 \cdot 2^{-2} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{4} \cdot 43^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(-0.21282441332723654392011193735\dots\)
Stable Faltings height:	\(-0.53714277150312209477100406126\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.70466600371099986663414003449\dots\)
Tamagawa product:	\( 2 \)  = \( 2\cdot1\cdot1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   490.2.a.e

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

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

Modular degree:	120
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 1.4093320074219997332682800689840508871 \)

Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(I_{2}\)	Split multiplicative	-1	1	2	2
\(5\)	\(1\)	\(I_{1}\)	Split multiplicative	-1	1	1	1
\(7\)	\(1\)	\(II\)	Additive	-1	2	2	0

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
\(7\)	B.1.3

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	split	ss	split	add	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	1	0,0	1	-	0	0,0	0	0	0	0	0	0	0	0	0
$\mu$-invariant(s)	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.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 490k consists of 2 curves linked by isogenies of degree 7.

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
$3$	3.1.980.1	\(\Z/2\Z\)	Not in database
$6$	6.0.19208000.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	\(\Q(\zeta_{7})\)	\(\Z/7\Z\)	Not in database
$7$	7.1.16807000000.2	\(\Z/7\Z\)	Not in database
$8$	8.2.2572983630000.1	\(\Z/3\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$18$	18.0.303843936636352000000.1	\(\Z/14\Z\)	Not in database
$21$	21.1.297767057903624960000000000000000000.1	\(\Z/14\Z\)	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.