Elliptic curve with Cremona label 97020ca2 (LMFDB label 97020.q1)

• Label: 97020ca2
• Conductor: $97020$
• Discriminant: $5.592\times 10^{16}$
• j-invariant: \( \frac{1711503051568}{7425} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-4884663x+4155261838\)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(1283, 414\right) \)
\(\hat{h}(P)\)	≈	$2.9905202715506295560678894151$

Torsion generators

\( \left(1274, 0\right) \)

Integral points

\((-441,\pm 78890)\), \( \left(1274, 0\right) \), \((1283,\pm 414)\)

Invariants

Conductor:	\( 97020 \)	=	\(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Discriminant:	\(55917315279302400 \)	=	\(2^{8} \cdot 3^{9} \cdot 5^{2} \cdot 7^{9} \cdot 11 \)
j-invariant:	\( \frac{1711503051568}{7425} \)	=	\(2^{4} \cdot 3^{-3} \cdot 5^{-2} \cdot 11^{-1} \cdot 47^{3} \cdot 101^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(2.9905202715506295560678894151\)
Real period:	\(0.31122344500742739115828831821\)
Tamagawa product:	\( 24 \)  = \( 3\cdot2\cdot2\cdot2\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 97020.2.a.q

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

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

Modular degree:	1806336
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 5.5843201276592051170911979440362138849 \)

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
\(3\)	\(2\)	\(I_3^{*}\)	Additive	-1	2	9	3
\(5\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(7\)	\(2\)	\(III^{*}\)	Additive	-1	2	9	0
\(11\)	\(1\)	\(I_{1}\)	Split multiplicative	-1	1	1	1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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

prime	Image of Galois representation
\(2\)	B

$p$-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	nonsplit	add	split	ordinary	ordinary	ordinary	ordinary	ordinary	ss	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	-	-	1	-	2	1	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	0,0

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 \(d=\) 2.
Its isogeny class 97020ca consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{231}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.0.4527600.7	\(\Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	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.