Properties

Label 64a
Number of curves $4$
Conductor $64$
CM \(\Q(\sqrt{-1}) \)
Rank $0$
Graph

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

Elliptic curves in class 64a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
64.a3 64a1 \([0, 0, 0, -4, 0]\) \(1728\) \(4096\) \([2, 2]\) \(2\) \(-0.61739\) \(\Gamma_0(N)\)-optimal \(-4\)
64.a1 64a2 \([0, 0, 0, -44, -112]\) \(287496\) \(32768\) \([2]\) \(4\) \(-0.27081\)   \(-16\)
64.a2 64a3 \([0, 0, 0, -44, 112]\) \(287496\) \(32768\) \([4]\) \(4\) \(-0.27081\)   \(-16\)
64.a4 64a4 \([0, 0, 0, 1, 0]\) \(1728\) \(-64\) \([2]\) \(4\) \(-0.96396\)   \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 64a have rank \(0\).

Complex multiplication

Each elliptic curve in class 64a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 64.2.a.a

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - 3q^{9} - 6q^{13} + 2q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.