Elliptic curve isogeny class with Cremona label 39710s (LMFDB label 39710.bc)

• Label: 39710s
• Number of curves: $4$
• Conductor: $39710$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 39710s have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(5\)	\(1 + T\)
\(11\)	\(1 + T\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + T + 3 T^{2}\)	1.3.b
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(13\)	\( 1 + 3 T + 13 T^{2}\)	1.13.d
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 + 4 T + 29 T^{2}\)	1.29.e
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 39710s do not have complex multiplication.

Modular form 39710.2.a.s

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 39710s

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
39710.bc4	39710s1	\([1, 1, 1, -569846, -165798557]\)	\(434985385981609/30179600\)	\(1419825870227600\)	\([2]\)	\(483840\)	\(1.9610\)	\(\Gamma_0(N)\)-optimal
39710.bc3	39710s2	\([1, 1, 1, -605946, -143647597]\)	\(523002686860009/113851032020\)	\(5356222104140109620\)	\([2]\)	\(967680\)	\(2.3075\)
39710.bc2	39710s3	\([1, 1, 1, -1152861, 224780339]\)	\(3601910963276569/1618496000000\)	\(76143570214976000000\)	\([2]\)	\(1451520\)	\(2.5103\)
39710.bc1	39710s4	\([1, 1, 1, -15592861, 23681116339]\)	\(8912089320684236569/5116268168000\)	\(240699343395816008000\)	\([2]\)	\(2903040\)	\(2.8568\)