Elliptic curve isogeny class with Cremona label 8330e (LMFDB label 8330.d)

• Label: 8330e
• Number of curves: $2$
• Conductor: $8330$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 8330e have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(5\)	\(1 + T\)
\(7\)	\(1\)
\(17\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + T + 3 T^{2}\)	1.3.b
\(11\)	\( 1 - 2 T + 11 T^{2}\)	1.11.ac
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + 5 T + 23 T^{2}\)	1.23.f
\(29\)	\( 1 - T + 29 T^{2}\)	1.29.ab
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 8330e do not have complex multiplication.

Modular form 8330.2.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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 8330e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
8330.d1	8330e1	\([1, 1, 0, -123, -2267]\)	\(-1771561/17000\)	\(-2000033000\)	\([]\)	\(4320\)	\(0.46705\)	\(\Gamma_0(N)\)-optimal
8330.d2	8330e2	\([1, 1, 0, 1102, 57268]\)	\(1256216039/12577280\)	\(-1479704414720\)	\([]\)	\(12960\)	\(1.0164\)