Properties

Label 236992cj
Number of curves $6$
Conductor $236992$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 236992cj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
236992.cj5 236992cj1 \([0, -1, 0, -17633, -1818271]\) \(-15625/28\) \(-1086588162408448\) \([2]\) \(811008\) \(1.5754\) \(\Gamma_0(N)\)-optimal
236992.cj4 236992cj2 \([0, -1, 0, -356193, -81650719]\) \(128787625/98\) \(3803058568429568\) \([2]\) \(1622016\) \(1.9220\)  
236992.cj6 236992cj3 \([0, -1, 0, 151647, 38097953]\) \(9938375/21952\) \(-851885119328223232\) \([2]\) \(2433024\) \(2.1247\)  
236992.cj3 236992cj4 \([0, -1, 0, -1202593, 416472609]\) \(4956477625/941192\) \(36524574491197571072\) \([2]\) \(4866048\) \(2.4713\)  
236992.cj2 236992cj5 \([0, -1, 0, -5773153, 5356435425]\) \(-548347731625/1835008\) \(-71210641811600048128\) \([2]\) \(7299072\) \(2.6740\)  
236992.cj1 236992cj6 \([0, -1, 0, -92444513, 342144006113]\) \(2251439055699625/25088\) \(973582993517969408\) \([2]\) \(14598144\) \(3.0206\)  

Rank

sage: E.rank()
 

The elliptic curves in class 236992cj have rank \(0\).

Complex multiplication

The elliptic curves in class 236992cj do not have complex multiplication.

Modular form 236992.2.a.cj

sage: E.q_eigenform(10)
 
\(q + 2q^{3} - q^{7} + q^{9} + 4q^{13} - 6q^{17} + 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.