Elliptic curve isogeny class with Cremona label 51408bc (LMFDB label 51408.br)

• Label: 51408bc
• Number of curves: $3$
• Conductor: $51408$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 51408bc have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + T + 5 T^{2}\)	1.5.b
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - T + 13 T^{2}\)	1.13.ab
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(23\)	\( 1 + 3 T + 23 T^{2}\)	1.23.d
\(29\)	\( 1 - 3 T + 29 T^{2}\)	1.29.ad
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 51408bc do not have complex multiplication.

Modular form 51408.2.a.bc

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 51408bc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
51408.br3	51408bc1	\([0, 0, 0, -315360, 67791152]\)	\(31363160518656000/198257271191\)	\(21925668135555072\)	\([]\)	\(357696\)	\(1.9728\)	\(\Gamma_0(N)\)-optimal
51408.br1	51408bc2	\([0, 0, 0, -25505280, 49578485616]\)	\(22759502184972288000/5831\)	\(470104363008\)	\([]\)	\(1073088\)	\(2.5221\)
51408.br2	51408bc3	\([0, 0, 0, -1961760, -1012313104]\)	\(838870874148864000/40675641638471\)	\(40485605040736063488\)	\([]\)	\(1073088\)	\(2.5221\)