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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1680.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.g1 | 1680m7 | \([0, -1, 0, -5619720, 5129544432]\) | \(4791901410190533590281/41160000\) | \(168591360000\) | \([4]\) | \(27648\) | \(2.1954\) | |
| 1680.g2 | 1680m6 | \([0, -1, 0, -351240, 80233200]\) | \(1169975873419524361/108425318400\) | \(444110104166400\) | \([2, 2]\) | \(13824\) | \(1.8488\) | |
| 1680.g3 | 1680m8 | \([0, -1, 0, -325640, 92398320]\) | \(-932348627918877961/358766164249920\) | \(-1469506208767672320\) | \([2]\) | \(27648\) | \(2.1954\) | |
| 1680.g4 | 1680m4 | \([0, -1, 0, -69720, 6984432]\) | \(9150443179640281/184570312500\) | \(756000000000000\) | \([4]\) | \(9216\) | \(1.6461\) | |
| 1680.g5 | 1680m3 | \([0, -1, 0, -23560, 1065712]\) | \(353108405631241/86318776320\) | \(353561707806720\) | \([2]\) | \(6912\) | \(1.5023\) | |
| 1680.g6 | 1680m2 | \([0, -1, 0, -9240, -176400]\) | \(21302308926361/8930250000\) | \(36578304000000\) | \([2, 2]\) | \(4608\) | \(1.2995\) | |
| 1680.g7 | 1680m1 | \([0, -1, 0, -7960, -270608]\) | \(13619385906841/6048000\) | \(24772608000\) | \([2]\) | \(2304\) | \(0.95296\) | \(\Gamma_0(N)\)-optimal |
| 1680.g8 | 1680m5 | \([0, -1, 0, 30760, -1328400]\) | \(785793873833639/637994920500\) | \(-2613227194368000\) | \([2]\) | \(9216\) | \(1.6461\) |
Rank
sage: E.rank()
The elliptic curves in class 1680.g have rank \(1\).
Complex multiplication
The elliptic curves in class 1680.g do not have complex multiplication.Modular form 1680.2.a.g
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
