Elliptic curve isogeny class with Cremona label 210210ej (LMFDB label 210210.bf)

• Label: 210210ej
• Number of curves: $4$
• Conductor: $210210$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 210210ej have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 - T\)
\(5\)	\(1 + T\)
\(7\)	\(1\)
\(11\)	\(1 + T\)
\(13\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 210210ej do not have complex multiplication.

Modular form 210210.2.a.ej

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 210210ej

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
210210.bf4	210210ej1	\([1, 1, 0, -1642, -321404]\)	\(-4165509529/375289200\)	\(-44152399090800\)	\([2]\)	\(589824\)	\(1.2978\)	\(\Gamma_0(N)\)-optimal
210210.bf3	210210ej2	\([1, 1, 0, -81022, -8846816]\)	\(499980107400409/4140922500\)	\(487175391202500\)	\([2, 2]\)	\(1179648\)	\(1.6444\)
210210.bf2	210210ej3	\([1, 1, 0, -138352, 5221966]\)	\(2489411558640889/1338278906250\)	\(157447175041406250\)	\([2]\)	\(2359296\)	\(1.9910\)
210210.bf1	210210ej4	\([1, 1, 0, -1293772, -566954366]\)	\(2035678735521204409/141376950\)	\(16632856790550\)	\([2]\)	\(2359296\)	\(1.9910\)