Elliptic curve isogeny class with Cremona label 5070k (LMFDB label 5070.k)

• Label: 5070k
• Number of curves: $4$
• Conductor: $5070$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 5070k have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(3\)	\(1 - T\)
\(5\)	\(1 - T\)
\(13\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 5070k do not have complex multiplication.

Modular form 5070.2.a.k

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 5070k

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
5070.k3	5070k1	\([1, 0, 1, -1018, 80108]\)	\(-24137569/561600\)	\(-2710735934400\)	\([2]\)	\(8064\)	\(1.0664\)	\(\Gamma_0(N)\)-optimal
5070.k2	5070k2	\([1, 0, 1, -34818, 2486668]\)	\(967068262369/4928040\)	\(23786707824360\)	\([2]\)	\(16128\)	\(1.4130\)
5070.k4	5070k3	\([1, 0, 1, 9122, -2118244]\)	\(17394111071/411937500\)	\(-1988343632437500\)	\([2]\)	\(24192\)	\(1.6157\)
5070.k1	5070k4	\([1, 0, 1, -202128, -33214244]\)	\(189208196468929/10860320250\)	\(52420691525582250\)	\([2]\)	\(48384\)	\(1.9623\)