Elliptic curve isogeny class with Cremona label 8470u (LMFDB label 8470.r)

• Label: 8470u
• Number of curves: $4$
• Conductor: $8470$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 8470u have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(5\)	\(1 - T\)
\(7\)	\(1 + T\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 2 T + 3 T^{2}\)	1.3.c
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(19\)	\( 1 - T + 19 T^{2}\)	1.19.ab
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 8470u do not have complex multiplication.

Modular form 8470.2.a.u

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 8470u

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
8470.r3	8470u1	\([1, 0, 0, -110536, -14214144]\)	\(-84309998289049/414124480\)	\(-733646777913280\)	\([2]\)	\(69120\)	\(1.7006\)	\(\Gamma_0(N)\)-optimal
8470.r2	8470u2	\([1, 0, 0, -1770656, -907026680]\)	\(346553430870203929/8300600\)	\(14705019236600\)	\([2]\)	\(138240\)	\(2.0472\)
8470.r4	8470u3	\([1, 0, 0, 274849, -75395495]\)	\(1296134247276791/2137096192000\)	\(-3785996266995712000\)	\([2]\)	\(207360\)	\(2.2499\)
8470.r1	8470u4	\([1, 0, 0, -1893471, -774028199]\)	\(423783056881319689/99207416000000\)	\(175751989096376000000\)	\([2]\)	\(414720\)	\(2.5965\)