Properties

Label 361998cr
Number of curves $4$
Conductor $361998$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 361998cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
361998.cr4 361998cr1 \([1, -1, 1, 1489, -5977929]\) \(103823/4386816\) \(-15436081430654976\) \([2]\) \(2949120\) \(1.7852\) \(\Gamma_0(N)\)-optimal
361998.cr3 361998cr2 \([1, -1, 1, -485231, -127657929]\) \(3590714269297/73410624\) \(258313175191116864\) \([2, 2]\) \(5898240\) \(2.1318\)  
361998.cr2 361998cr3 \([1, -1, 1, -1032791, 213800487]\) \(34623662831857/14438442312\) \(50805178803908415432\) \([2]\) \(11796480\) \(2.4783\)  
361998.cr1 361998cr4 \([1, -1, 1, -7725191, -8262476985]\) \(14489843500598257/6246072\) \(21978326880756792\) \([2]\) \(11796480\) \(2.4783\)  

Rank

sage: E.rank()
 

The elliptic curves in class 361998cr have rank \(0\).

Complex multiplication

The elliptic curves in class 361998cr do not have complex multiplication.

Modular form 361998.2.a.cr

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2q^{5} + q^{7} + q^{8} - 2q^{10} + q^{14} + q^{16} - q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.