Elliptic curve isogeny class with Cremona label 31850ce (LMFDB label 31850.bo)

• Label: 31850ce
• Number of curves: $4$
• Conductor: $31850$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 31850ce have rank \(2\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + T + 23 T^{2}\)	1.23.b
\(29\)	\( 1 - 3 T + 29 T^{2}\)	1.29.ad
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 31850ce do not have complex multiplication.

Modular form 31850.2.a.ce

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 31850ce

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
31850.bo2	31850ce1	\([1, 0, 0, -39838, -3045708]\)	\(3803721481/26000\)	\(47794906250000\)	\([2]\)	\(165888\)	\(1.4588\)	\(\Gamma_0(N)\)-optimal
31850.bo3	31850ce2	\([1, 0, 0, -15338, -6745208]\)	\(-217081801/10562500\)	\(-19416680664062500\)	\([2]\)	\(331776\)	\(1.8054\)
31850.bo1	31850ce3	\([1, 0, 0, -254213, 47332417]\)	\(988345570681/44994560\)	\(82711952960000000\)	\([2]\)	\(497664\)	\(2.0081\)
31850.bo4	31850ce4	\([1, 0, 0, 137787, 180220417]\)	\(157376536199/7722894400\)	\(-14196731301025000000\)	\([2]\)	\(995328\)	\(2.3547\)