Properties

Label 330.a
Number of curves $4$
Conductor $330$
CM no
Rank $0$
Graph

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

Elliptic curves in class 330.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330.a1 330a3 \([1, 1, 0, -4163, 77343]\) \(7981893677157049/1917731420550\) \(1917731420550\) \([2]\) \(640\) \(1.0680\)  
330.a2 330a2 \([1, 1, 0, -1413, -20007]\) \(312341975961049/17862322500\) \(17862322500\) \([2, 2]\) \(320\) \(0.72141\)  
330.a3 330a1 \([1, 1, 0, -1393, -20603]\) \(299270638153369/1069200\) \(1069200\) \([2]\) \(160\) \(0.37484\) \(\Gamma_0(N)\)-optimal
330.a4 330a4 \([1, 1, 0, 1017, -78813]\) \(116149984977671/2779502343750\) \(-2779502343750\) \([2]\) \(640\) \(1.0680\)  

Rank

sage: E.rank()
 

The elliptic curves in class 330.a have rank \(0\).

Complex multiplication

The elliptic curves in class 330.a do not have complex multiplication.

Modular form 330.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 2 q^{13} + q^{15} + q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.