Elliptic curve with Cremona label 97020bf1 (LMFDB label 97020.v2)

• Label: 97020bf1
• Conductor: $97020$
• Discriminant: $-1.627\times 10^{16}$
• j-invariant: \( \frac{16674224}{15125} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3+54537x-3692738\)

Mordell-Weil group structure

trivial

Integral points

None

Invariants

Conductor:	\( 97020 \)	=	\(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Discriminant:	\(-16272234605088000 \)	=	\(-1 \cdot 2^{8} \cdot 3^{6} \cdot 5^{3} \cdot 7^{8} \cdot 11^{2} \)
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:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.21468321121927945061645083105\)
Tamagawa product:	\( 6 \)  = \( 1\cdot1\cdot1\cdot3\cdot2 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 97020.2.a.v

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

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

Modular degree:	725760
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 1.2880992673156767036987049863280697308 \)

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\((j)_{-}\)
\(2\)	\(1\)	\(IV^{*}\)	Additive	-1	2	8	0
\(3\)	\(1\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(5\)	\(1\)	\(I_{3}\)	Non-split multiplicative	1	1	3	3
\(7\)	\(3\)	\(IV^{*}\)	Additive	1	2	8	0
\(11\)	\(2\)	\(I_{2}\)	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.1.2

$p$-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	add	add	nonsplit	add	split	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	-	0	-	1	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

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 97020bf 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
$2$	\(\Q(\sqrt{-3}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.980.1	\(\Z/2\Z\)	Not in database
$3$	3.1.71148.1	\(\Z/3\Z\)	Not in database
$6$	6.0.19208000.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.0.15186113712.1	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$6$	6.0.25930800.1	\(\Z/6\Z\)	Not in database
$9$	9.1.720307745587584000.1	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$18$	18.0.58399516821105745300323000000000000.1	\(\Z/9\Z\)	Not in database
$18$	18.0.14008767705543626210378203840512000000.1	\(\Z/3\Z \times \Z/6\Z\)	Not in database
$18$	18.0.1037686496706935274842829914112000000000.1	\(\Z/2\Z \times \Z/6\Z\)	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.