Elliptic curve isogeny class with Cremona label 47040es (LMFDB label 47040.b)

• Label: 47040es
• Number of curves: $2$
• Conductor: $47040$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 47040es have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.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 47040es do not have complex multiplication.

Modular form 47040.2.a.es

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.

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

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 47040es

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
47040.b1	47040es1	\([0, -1, 0, -167841, 26479041]\)	\(197723452/375\)	\(991730245632000\)	\([2]\)	\(430080\)	\(1.7673\)	\(\Gamma_0(N)\)-optimal
47040.b2	47040es2	\([0, -1, 0, -112961, 44029665]\)	\(-30138446/140625\)	\(-743797684224000000\)	\([2]\)	\(860160\)	\(2.1139\)