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Properties

Label 286110cl
Number of curves $6$
Conductor $286110$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 286110cl have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 286110cl do not have complex multiplication.

Modular form 286110.2.a.cl

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - q^{11} - 2 q^{13} + q^{16} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 286110cl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
286110.cl5 286110cl1 \([1, -1, 0, 12951, -2203187]\) \(13651919/126720\) \(-2229801590142720\) \([2]\) \(1310720\) \(1.6263\) \(\Gamma_0(N)\)-optimal
286110.cl4 286110cl2 \([1, -1, 0, -195129, -30626915]\) \(46694890801/3920400\) \(68984486695040400\) \([2, 2]\) \(2621440\) \(1.9729\)  
286110.cl3 286110cl3 \([1, -1, 0, -663309, 172844113]\) \(1834216913521/329422500\) \(5796613118124922500\) \([2, 2]\) \(5242880\) \(2.3194\)  
286110.cl2 286110cl4 \([1, -1, 0, -3056229, -2055713495]\) \(179415687049201/1443420\) \(25398833737719420\) \([2]\) \(5242880\) \(2.3194\)  
286110.cl1 286110cl5 \([1, -1, 0, -10104939, 12365765095]\) \(6484907238722641/283593750\) \(4990197243564843750\) \([2]\) \(10485760\) \(2.6660\)  
286110.cl6 286110cl6 \([1, -1, 0, 1287441, 996450763]\) \(13411719834479/32153832150\) \(-565788084416446602150\) \([2]\) \(10485760\) \(2.6660\)