Elliptic curve isogeny class with Cremona label 68400fa (LMFDB label 68400.ds)

• Label: 68400fa
• Number of curves: $4$
• Conductor: $68400$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 68400fa have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1\)
\(19\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 - 2 T + 7 T^{2}\)	1.7.ac
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 68400fa do not have complex multiplication.

Modular form 68400.2.a.fa

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 68400fa

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
68400.ds3	68400fa1	\([0, 0, 0, -111675, 14278250]\)	\(3301293169/22800\)	\(1063756800000000\)	\([2]\)	\(294912\)	\(1.7169\)	\(\Gamma_0(N)\)-optimal
68400.ds2	68400fa2	\([0, 0, 0, -183675, -6385750]\)	\(14688124849/8122500\)	\(378963360000000000\)	\([2, 2]\)	\(589824\)	\(2.0635\)
68400.ds4	68400fa3	\([0, 0, 0, 716325, -50485750]\)	\(871257511151/527800050\)	\(-24625039132800000000\)	\([2]\)	\(1179648\)	\(2.4101\)
68400.ds1	68400fa4	\([0, 0, 0, -2235675, -1284781750]\)	\(26487576322129/44531250\)	\(2077650000000000000\)	\([2]\)	\(1179648\)	\(2.4101\)