Properties

Label 212415t
Number of curves 4
Conductor 212415
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("212415.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212415t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
212415.w4 212415t1 [1, 0, 0, 197959, -57000000] [2] 3538944 \(\Gamma_0(N)\)-optimal
212415.w3 212415t2 [1, 0, 0, -1572166, -622377925] [2, 2] 7077888  
212415.w2 212415t3 [1, 0, 0, -7590591, 7482033180] [2] 14155776  
212415.w1 212415t4 [1, 0, 0, -23875741, -44903895730] [2] 14155776  

Rank

sage: E.rank()
 

The elliptic curves in class 212415t have rank \(1\).

Modular form 212415.2.a.w

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

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.