Elliptic curve isogeny class with LMFDB label 3213.k (Cremona label 3213r)

• Label: 3213.k
• Number of curves: $3$
• Conductor: $3213$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 3213.k have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)
\(7\)	\(1 - T\)
\(17\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(19\)	\( 1 - 2 T + 19 T^{2}\)	1.19.ac
\(23\)	\( 1 - 9 T + 23 T^{2}\)	1.23.aj
\(29\)	\( 1 + 9 T + 29 T^{2}\)	1.29.j
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 3213.k do not have complex multiplication.

Modular form 3213.2.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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 3213.k

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3213.k1	3213r3	\([0, 0, 1, -1103490, -427069591]\)	\(838870874148864000/40675641638471\)	\(7205567889330222237\)	\([]\)	\(44712\)	\(2.3782\)
3213.k2	3213r2	\([0, 0, 1, -177390, 28599392]\)	\(31363160518656000/198257271191\)	\(3902297868852453\)	\([3]\)	\(14904\)	\(1.8289\)
3213.k3	3213r1	\([0, 0, 1, -177120, 28691253]\)	\(22759502184972288000/5831\)	\(157437\)	\([3]\)	\(4968\)	\(1.2796\)	\(\Gamma_0(N)\)-optimal