Properties

Label 576.c
Number of curves $4$
Conductor $576$
CM \(\Q(\sqrt{-1}) \)
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 576.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
576.c1 576h3 \([0, 0, 0, -396, -3024]\) \(287496\) \(23887872\) \([2]\) \(128\) \(0.27849\)   \(-16\)
576.c2 576h4 \([0, 0, 0, -396, 3024]\) \(287496\) \(23887872\) \([2]\) \(128\) \(0.27849\)   \(-16\)
576.c3 576h2 \([0, 0, 0, -36, 0]\) \(1728\) \(2985984\) \([2, 2]\) \(64\) \(-0.068080\)   \(-4\)
576.c4 576h1 \([0, 0, 0, 9, 0]\) \(1728\) \(-46656\) \([2]\) \(32\) \(-0.41465\) \(\Gamma_0(N)\)-optimal \(-4\)

Rank

sage: E.rank()
 

The elliptic curves in class 576.c have rank \(1\).

Complex multiplication

Each elliptic curve in class 576.c has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 576.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.