Elliptic curve isogeny class with Cremona label 159936df (LMFDB label 159936.bn)

• Label: 159936df
• Number of curves: $4$
• Conductor: $159936$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 159936df have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 + T\)
\(7\)	\(1\)
\(17\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - T + 5 T^{2}\)	1.5.ab
\(11\)	\( 1 + 2 T + 11 T^{2}\)	1.11.c
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(19\)	\( 1 + 7 T + 19 T^{2}\)	1.19.h
\(23\)	\( 1 + 5 T + 23 T^{2}\)	1.23.f
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 159936df do not have complex multiplication.

Modular form 159936.2.a.df

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 159936df

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
159936.bn3	159936df1	\([0, -1, 0, -6488204, 6363229650]\)	\(4011705594213827392/52680152007\)	\(396657101022178752\)	\([2]\)	\(3538944\)	\(2.5209\)	\(\Gamma_0(N)\)-optimal
159936.bn2	159936df2	\([0, -1, 0, -6666809, 5994553209]\)	\(68003243639904448/7163272192041\)	\(3451911414257383870464\)	\([2, 2]\)	\(7077888\)	\(2.8675\)
159936.bn4	159936df3	\([0, -1, 0, 8627071, 29470659009]\)	\(18419405270942584/108003564029403\)	\(-416366962425699812868096\)	\([2]\)	\(14155776\)	\(3.2141\)
159936.bn1	159936df4	\([0, -1, 0, -24818369, -41079702495]\)	\(438536015880092936/64602489661101\)	\(249050507055558542917632\)	\([2]\)	\(14155776\)	\(3.2141\)