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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 10816.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
10816.r1 | 10816bb3 | \([0, 0, 0, -7436, -246064]\) | \(287496\) | \(158164877312\) | \([2]\) | \(7680\) | \(1.0117\) | \(-16\) | |
10816.r2 | 10816bb4 | \([0, 0, 0, -7436, 246064]\) | \(287496\) | \(158164877312\) | \([2]\) | \(7680\) | \(1.0117\) | \(-16\) | |
10816.r3 | 10816bb2 | \([0, 0, 0, -676, 0]\) | \(1728\) | \(19770609664\) | \([2, 2]\) | \(3840\) | \(0.66509\) | \(-4\) | |
10816.r4 | 10816bb1 | \([0, 0, 0, 169, 0]\) | \(1728\) | \(-308915776\) | \([2]\) | \(1920\) | \(0.31852\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 10816.r have rank \(0\).
Complex multiplication
Each elliptic curve in class 10816.r has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 10816.2.a.r
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.