Elliptic curve isogeny class with Cremona label 47040e (LMFDB label 47040.r)

• Label: 47040e
• Number of curves: $4$
• Conductor: $47040$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 47040e have rank \(2\).

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 \(\mathbb{F}_p\)
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 47040e do not have complex multiplication.

Modular form 47040.2.a.e

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.

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

Isogeny graph

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

Elliptic curves in class 47040e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
47040.r4	47040e1	\([0, -1, 0, 1699, 221901]\)	\(4499456/180075\)	\(-21694099123200\)	\([2]\)	\(98304\)	\(1.2389\)	\(\Gamma_0(N)\)-optimal
47040.r3	47040e2	\([0, -1, 0, -46321, 3688945]\)	\(5702413264/275625\)	\(531284060160000\)	\([2, 2]\)	\(196608\)	\(1.5854\)
47040.r2	47040e3	\([0, -1, 0, -128641, -12956159]\)	\(30534944836/8203125\)	\(63248102400000000\)	\([2]\)	\(393216\)	\(1.9320\)
47040.r1	47040e4	\([0, -1, 0, -732321, 241456545]\)	\(5633270409316/14175\)	\(109292720947200\)	\([2]\)	\(393216\)	\(1.9320\)