Elliptic curve with Cremona label 97020a1 (LMFDB label 97020.l1)

• Label: 97020a1
• Conductor: $97020$
• Discriminant: $-1.205\times 10^{14}$
• j-invariant: \( -\frac{1647408715474378752}{3025} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{3}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-84040488x+296538415012\)

Mordell-Weil group structure

\(\Z/{3}\Z\)

Torsion generators

\( \left(5292, 98\right) \)

Integral points

\((5292,\pm 98)\)

Invariants

Conductor:	\( 97020 \)	=	\(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Discriminant:	\(-120535071148800 \)	=	\(-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{2} \cdot 7^{8} \cdot 11^{2} \)
j-invariant:	\( -\frac{1647408715474378752}{3025} \)	=	\(-1 \cdot 2^{13} \cdot 3^{6} \cdot 5^{-2} \cdot 7 \cdot 11^{-2} \cdot 41^{3} \cdot 83^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.26927441170297089433384327295\)
Tamagawa product:	\( 72 \)  = \( 3\cdot2\cdot2\cdot3\cdot2 \)
Torsion order:	\(3\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 97020.2.a.l

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

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

Modular degree:	4499712
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 2.1541952936237671546707461836246971130 \)

Local data

This elliptic curve is not semistable. There are 5 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
\(3\)	\(2\)	\(III\)	Additive	1	2	3	0
\(5\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(7\)	\(3\)	\(IV^{*}\)	Additive	1	2	8	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.1.1

$p$-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
Reduction type	add	add	nonsplit	add	nonsplit
$\lambda$-invariant(s)	-	-	0	-	0
$\mu$-invariant(s)	-	-	0	-	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 97020a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.588.1	\(\Z/6\Z\)	Not in database
$6$	6.0.1037232.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$6$	6.0.768797006670000.1	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$9$	9.3.988639123509326520000.3	\(\Z/9\Z\)	Not in database
$12$	Deg 12	\(\Z/12\Z\)	Not in database
$18$	18.0.454396577038684708060888856640963000000000000.2	\(\Z/6\Z \times \Z/6\Z\)	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.