Elliptic curve isogeny class with Cremona label 57330ez (LMFDB label 57330.fg)

• Label: 57330ez
• Number of curves: $4$
• Conductor: $57330$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 57330ez have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1\)
\(5\)	\(1 - T\)
\(7\)	\(1\)
\(13\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 - 2 T + 11 T^{2}\)	1.11.ac
\(17\)	\( 1 + 4 T + 17 T^{2}\)	1.17.e
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 + 4 T + 29 T^{2}\)	1.29.e
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 57330ez do not have complex multiplication.

Modular form 57330.2.a.ez

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 57330ez

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
57330.fg2	57330ez1	\([1, -1, 1, -14342, 660741]\)	\(3803721481/26000\)	\(2229919146000\)	\([2]\)	\(165888\)	\(1.2034\)	\(\Gamma_0(N)\)-optimal
57330.fg3	57330ez2	\([1, -1, 1, -5522, 1458069]\)	\(-217081801/10562500\)	\(-905904653062500\)	\([2]\)	\(331776\)	\(1.5500\)
57330.fg1	57330ez3	\([1, -1, 1, -91517, -10205499]\)	\(988345570681/44994560\)	\(3859008877301760\)	\([2]\)	\(497664\)	\(1.7527\)
57330.fg4	57330ez4	\([1, -1, 1, 49603, -38937531]\)	\(157376536199/7722894400\)	\(-662362695580622400\)	\([2]\)	\(995328\)	\(2.0993\)