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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 7920d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7920.h5 | 7920d1 | \([0, 0, 0, -1578, 40327]\) | \(-37256083456/38671875\) | \(-451068750000\) | \([2]\) | \(8192\) | \(0.93155\) | \(\Gamma_0(N)\)-optimal |
| 7920.h4 | 7920d2 | \([0, 0, 0, -29703, 1969702]\) | \(15529488955216/6125625\) | \(1143188640000\) | \([2, 2]\) | \(16384\) | \(1.2781\) | |
| 7920.h3 | 7920d3 | \([0, 0, 0, -34203, 1333402]\) | \(5927735656804/2401490025\) | \(1792702697702400\) | \([2, 2]\) | \(32768\) | \(1.6247\) | |
| 7920.h1 | 7920d4 | \([0, 0, 0, -475203, 126086002]\) | \(15897679904620804/2475\) | \(1847577600\) | \([2]\) | \(32768\) | \(1.6247\) | |
| 7920.h2 | 7920d5 | \([0, 0, 0, -252003, -47758718]\) | \(1185450336504002/26043266205\) | \(38882388097935360\) | \([2]\) | \(65536\) | \(1.9713\) | |
| 7920.h6 | 7920d6 | \([0, 0, 0, 111597, 9702322]\) | \(102949393183198/86815346805\) | \(-129614618257090560\) | \([2]\) | \(65536\) | \(1.9713\) |
Rank
sage: E.rank()
The elliptic curves in class 7920d have rank \(0\).
Complex multiplication
The elliptic curves in class 7920d do not have complex multiplication.Modular form 7920.2.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.
