Elliptic curve isogeny class with LMFDB label 51744.bu (Cremona label 51744cr)

• Label: 51744.bu
• Number of curves: $4$
• Conductor: $51744$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 51744.bu have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(7\)	\(1\)
\(11\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 51744.bu do not have complex multiplication.

Modular form 51744.2.a.bu

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 LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 51744.bu

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
51744.bu1	51744cr4	\([0, 1, 0, -353649, -80932545]\)	\(10150654719808/19370043\)	\(9334235909763072\)	\([2]\)	\(442368\)	\(1.9539\)
51744.bu2	51744cr3	\([0, 1, 0, -293624, 60815292]\)	\(46477380430664/286446699\)	\(17254485857613312\)	\([2]\)	\(442368\)	\(1.9539\)
51744.bu3	51744cr1	\([0, 1, 0, -29514, -352584]\)	\(377619516352/211789809\)	\(1594678991298624\)	\([2, 2]\)	\(221184\)	\(1.6073\)	\(\Gamma_0(N)\)-optimal
51744.bu4	51744cr2	\([0, 1, 0, 116016, -2681064]\)	\(2866919053816/1712145897\)	\(-103133313349710336\)	\([2]\)	\(442368\)	\(1.9539\)