Elliptic curve isogeny class with Cremona label 73997j (LMFDB label 73997.c)

• Label: 73997j
• Number of curves: $3$
• Conductor: $73997$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 73997j have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(7\)	\(1 + T\)
\(11\)	\(1 + T\)
\(31\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(3\)	\( 1 - 3 T + 3 T^{2}\)	1.3.ad
\(5\)	\( 1 + 4 T + 5 T^{2}\)	1.5.e
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 3 T + 19 T^{2}\)	1.19.ad
\(23\)	\( 1 - 6 T + 23 T^{2}\)	1.23.ag
\(29\)	\( 1 + 9 T + 29 T^{2}\)	1.29.j
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 73997j do not have complex multiplication.

Modular form 73997.2.a.j

Isogeny matrix

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 73997j

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
73997.c1	73997j1	\([0, -1, 1, -85849, -9653183]\)	\(-78843215872/539\)	\(-478364484059\)	\([]\)	\(199800\)	\(1.4220\)	\(\Gamma_0(N)\)-optimal
73997.c2	73997j2	\([0, -1, 1, -47409, -18355038]\)	\(-13278380032/156590819\)	\(-138974928273304739\)	\([]\)	\(599400\)	\(1.9713\)
73997.c3	73997j3	\([0, -1, 1, 423481, 475373127]\)	\(9463555063808/115539436859\)	\(-102541675513029577979\)	\([]\)	\(1798200\)	\(2.5206\)