Properties

Label 315.b
Number of curves 3
Conductor 315
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("315.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 315.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
315.b1 315a3 [0, 0, 1, -1182, 16362] [3] 180  
315.b2 315a1 [0, 0, 1, -12, -18] [] 20 \(\Gamma_0(N)\)-optimal
315.b3 315a2 [0, 0, 1, 78, 45] [3] 60  

Rank

sage: E.rank()
 

The elliptic curves in class 315.b have rank \(0\).

Modular form 315.2.a.b

sage: E.q_eigenform(10)
 
\( q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.