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SageMath
E = EllipticCurve("a1")
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)
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.