Elliptic curve isogeny class with Cremona label 1850a (LMFDB label 1850.f)

• Label: 1850a
• Number of curves: $4$
• Conductor: $1850$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 1850a have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(5\)	\(1\)
\(37\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 2 T + 3 T^{2}\)	1.3.c
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1850a do not have complex multiplication.

Modular form 1850.2.a.a

Isogeny matrix

The \((i,j)\)-th 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 1850a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1850.f3	1850a1	\([1, 1, 0, -1875, -17875]\)	\(46694890801/18944000\)	\(296000000000\)	\([2]\)	\(2304\)	\(0.89892\)	\(\Gamma_0(N)\)-optimal
1850.f4	1850a2	\([1, 1, 0, 6125, -121875]\)	\(1625964918479/1369000000\)	\(-21390625000000\)	\([2]\)	\(4608\)	\(1.2455\)
1850.f1	1850a3	\([1, 1, 0, -131875, -18487875]\)	\(16232905099479601/4052240\)	\(63316250000\)	\([2]\)	\(6912\)	\(1.4482\)
1850.f2	1850a4	\([1, 1, 0, -131375, -18634375]\)	\(-16048965315233521/256572640900\)	\(-4008947514062500\)	\([2]\)	\(13824\)	\(1.7948\)