Properties

Label 212415o
Number of curves $6$
Conductor $212415$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 212415o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212415.bc4 212415o1 \([1, 0, 0, -55299000, -158283871425]\) \(6585576176607121/187425\) \(532242178301041425\) \([2]\) \(14155776\) \(2.9110\) \(\Gamma_0(N)\)-optimal
212415.bc3 212415o2 \([1, 0, 0, -55369805, -157858234248]\) \(6610905152742241/35128130625\) \(99755490268072689080625\) \([2, 2]\) \(28311552\) \(3.2576\)  
212415.bc2 212415o3 \([1, 0, 0, -86594810, 40251932475]\) \(25288177725059761/14387797265625\) \(40857903468640883141015625\) \([2, 2]\) \(56623104\) \(3.6042\)  
212415.bc5 212415o4 \([1, 0, 0, -25277680, -328727338423]\) \(-629004249876241/16074715228425\) \(-45648347065470693282562425\) \([2]\) \(56623104\) \(3.6042\)  
212415.bc1 212415o5 \([1, 0, 0, -1015910435, 12438995138100]\) \(40832710302042509761/91556816413125\) \(259999463284141348633963125\) \([2]\) \(113246208\) \(3.9508\)  
212415.bc6 212415o6 \([1, 0, 0, 343120735, 320684297142]\) \(1573196002879828319/926055908203125\) \(-2629777317916929473876953125\) \([2]\) \(113246208\) \(3.9508\)  

Rank

sage: E.rank()
 

The elliptic curves in class 212415o have rank \(0\).

Complex multiplication

The elliptic curves in class 212415o do not have complex multiplication.

Modular form 212415.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.