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SageMath
E = EllipticCurve("c1")
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)
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.