Properties

Label 315a
Number of curves $3$
Conductor $315$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 315a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
315.b2 315a1 \([0, 0, 1, -12, -18]\) \(-262144/35\) \(-25515\) \([]\) \(20\) \(-0.42184\) \(\Gamma_0(N)\)-optimal
315.b3 315a2 \([0, 0, 1, 78, 45]\) \(71991296/42875\) \(-31255875\) \([3]\) \(60\) \(0.12746\)  
315.b1 315a3 \([0, 0, 1, -1182, 16362]\) \(-250523582464/13671875\) \(-9966796875\) \([3]\) \(180\) \(0.67677\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 315a do not have complex multiplication.

Modular form 315.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.