Elliptic curve isogeny class with Cremona label 123969a (LMFDB label 123969.c)

• Label: 123969a
• Number of curves: $4$
• Conductor: $123969$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 123969a have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1 + T\)
\(31\)	\(1\)
\(43\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T + 2 T^{2}\)	1.2.b
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 + T + 13 T^{2}\)	1.13.b
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 5 T + 19 T^{2}\)	1.19.f
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 5 T + 29 T^{2}\)	1.29.af
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 123969a do not have complex multiplication.

Modular form 123969.2.a.a

Isogeny matrix

The \((i,j)\)-th 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 123969a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
123969.c3	123969a1	\([1, 1, 0, -23564, 1381635]\)	\(1630532233/1161\)	\(1030391773641\)	\([2]\)	\(216000\)	\(1.2407\)	\(\Gamma_0(N)\)-optimal
123969.c2	123969a2	\([1, 1, 0, -28369, 771400]\)	\(2845178713/1347921\)	\(1196284849197201\)	\([2, 2]\)	\(432000\)	\(1.5872\)
123969.c4	123969a3	\([1, 1, 0, 101366, 5986747]\)	\(129784785047/92307627\)	\(-81923358746874987\)	\([2]\)	\(864000\)	\(1.9338\)
123969.c1	123969a4	\([1, 1, 0, -234984, -43402887]\)	\(1616855892553/22851963\)	\(20281201280575803\)	\([2]\)	\(864000\)	\(1.9338\)