Elliptic curve isogeny class with LMFDB label 248004.e (Cremona label 248004e)

• Label: 248004.e
• Number of curves: $4$
• Conductor: $248004$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $1$

Rank

The elliptic curves in class 248004.e have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(83\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 248004.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 248004.2.a.e

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 248004.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
248004.e1	248004e4	\([0, 0, 0, -930015, 339641478]\)	\(54000\)	\(1647402846469638912\)	\([2]\)	\(3471552\)	\(2.2899\)		\(-12\)
248004.e2	248004e2	\([0, 0, 0, -103335, -12579314]\)	\(54000\)	\(2259811860726528\)	\([2]\)	\(1157184\)	\(1.7406\)		\(-12\)
248004.e3	248004e1	\([0, 0, 0, 0, -571787]\)	\(0\)	\(-141238241295408\)	\([2]\)	\(578592\)	\(1.3940\)	\(\Gamma_0(N)\)-optimal	\(-3\)
248004.e4	248004e3	\([0, 0, 0, 0, 15438249]\)	\(0\)	\(-102962677904352432\)	\([2]\)	\(1735776\)	\(1.9433\)		\(-3\)