Elliptic curve isogeny class with LMFDB label 76608.eg (Cremona label 76608bt)

• Label: 76608.eg
• Number of curves: $4$
• Conductor: $76608$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 76608.eg have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1 + T\)
\(19\)	\(1 - T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 76608.eg do not have complex multiplication.

Modular form 76608.2.a.eg

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 76608.eg

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
76608.eg1	76608bt4	\([0, 0, 0, -3617004, -2647704400]\)	\(27384399945278713/153257496\)	\(29287963579908096\)	\([2]\)	\(1179648\)	\(2.3519\)
76608.eg2	76608bt2	\([0, 0, 0, -230124, -39806800]\)	\(7052482298233/499254336\)	\(95408989390503936\)	\([2, 2]\)	\(589824\)	\(2.0053\)
76608.eg3	76608bt1	\([0, 0, 0, -45804, 3029168]\)	\(55611739513/11440128\)	\(2186242506620928\)	\([2]\)	\(294912\)	\(1.6588\)	\(\Gamma_0(N)\)-optimal
76608.eg4	76608bt3	\([0, 0, 0, 207636, -173411152]\)	\(5180411077127/70976229912\)	\(-13563768761443418112\)	\([2]\)	\(1179648\)	\(2.3519\)