Elliptic curve isogeny class with Cremona label 7728i (LMFDB label 7728.d)

• Label: 7728i
• Number of curves: $2$
• Conductor: $7728$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 7728i have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 + T\)
\(7\)	\(1 + T\)
\(23\)	\(1 - T\)

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
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 7728i do not have complex multiplication.

Modular form 7728.2.a.i

Isogeny matrix

The \((i,j)\)-th 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 7728i

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
7728.d1	7728i1	\([0, -1, 0, -478493, -127241151]\)	\(-47327266415721472000/1222082060283\)	\(-312853007432448\)	\([]\)	\(47520\)	\(1.8882\)	\(\Gamma_0(N)\)-optimal
7728.d2	7728i2	\([0, -1, 0, -148013, -299020767]\)	\(-1400832679220224000/150124273180279587\)	\(-38431813934151574272\)	\([]\)	\(142560\)	\(2.4375\)