Elliptic curve isogeny class with Cremona label 123200fn (LMFDB label 123200.hf)

• Label: 123200fn
• Number of curves: $4$
• Conductor: $123200$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 123200fn have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(5\)	\(1\)
\(7\)	\(1 + T\)
\(11\)	\(1 + T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 123200fn do not have complex multiplication.

Modular form 123200.2.a.fn

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 123200fn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
123200.hf3	123200fn1	\([0, -1, 0, -1461633, -682544863]\)	\(-84309998289049/414124480\)	\(-1696253870080000000\)	\([2]\)	\(2654208\)	\(2.3461\)	\(\Gamma_0(N)\)-optimal
123200.hf2	123200fn2	\([0, -1, 0, -23413633, -43598704863]\)	\(346553430870203929/8300600\)	\(33999257600000000\)	\([2]\)	\(5308416\)	\(2.6927\)
123200.hf4	123200fn3	\([0, -1, 0, 3634367, -3627640863]\)	\(1296134247276791/2137096192000\)	\(-8753546002432000000000\)	\([2]\)	\(7962624\)	\(2.8954\)
123200.hf1	123200fn4	\([0, -1, 0, -25037633, -37202552863]\)	\(423783056881319689/99207416000000\)	\(406353575936000000000000\)	\([2]\)	\(15925248\)	\(3.2420\)