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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 600d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 600.h5 | 600d1 | \([0, 1, 0, 17, 38]\) | \(2048/3\) | \(-750000\) | \([2]\) | \(64\) | \(-0.18721\) | \(\Gamma_0(N)\)-optimal |
| 600.h4 | 600d2 | \([0, 1, 0, -108, 288]\) | \(35152/9\) | \(36000000\) | \([2, 2]\) | \(128\) | \(0.15937\) | |
| 600.h3 | 600d3 | \([0, 1, 0, -608, -5712]\) | \(1556068/81\) | \(1296000000\) | \([2, 2]\) | \(256\) | \(0.50594\) | |
| 600.h2 | 600d4 | \([0, 1, 0, -1608, 24288]\) | \(28756228/3\) | \(48000000\) | \([2]\) | \(256\) | \(0.50594\) | |
| 600.h1 | 600d5 | \([0, 1, 0, -9608, -365712]\) | \(3065617154/9\) | \(288000000\) | \([2]\) | \(512\) | \(0.85251\) | |
| 600.h6 | 600d6 | \([0, 1, 0, 392, -21712]\) | \(207646/6561\) | \(-209952000000\) | \([2]\) | \(512\) | \(0.85251\) |
Rank
sage: E.rank()
The elliptic curves in class 600d have rank \(0\).
Complex multiplication
The elliptic curves in class 600d do not have complex multiplication.Modular form 600.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
