Elliptic curve isogeny class with Cremona label 27744f (LMFDB label 27744.d)

• Label: 27744f
• Number of curves: $4$
• Conductor: $27744$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 27744f have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + T + 5 T^{2}\)	1.5.b
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 - T + 11 T^{2}\)	1.11.ab
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 - 2 T + 23 T^{2}\)	1.23.ac
\(29\)	\( 1 + 7 T + 29 T^{2}\)	1.29.h
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 27744f do not have complex multiplication.

Modular form 27744.2.a.f

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 27744f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
27744.d3	27744f1	\([0, -1, 0, -674, 3864]\)	\(21952/9\)	\(13903239744\)	\([2, 2]\)	\(18432\)	\(0.64381\)	\(\Gamma_0(N)\)-optimal
27744.d4	27744f2	\([0, -1, 0, 2216, 25828]\)	\(97336/81\)	\(-1001033261568\)	\([2]\)	\(36864\)	\(0.99039\)
27744.d2	27744f3	\([0, -1, 0, -5009, -132255]\)	\(140608/3\)	\(296602447872\)	\([2]\)	\(36864\)	\(0.99039\)
27744.d1	27744f4	\([0, -1, 0, -9344, 350664]\)	\(7301384/3\)	\(37075305984\)	\([2]\)	\(36864\)	\(0.99039\)