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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 8400.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.cm1 | 8400cf7 | \([0, 1, 0, -140493008, 640912067988]\) | \(4791901410190533590281/41160000\) | \(2634240000000000\) | \([2]\) | \(663552\) | \(3.0001\) | |
| 8400.cm2 | 8400cf6 | \([0, 1, 0, -8781008, 10011587988]\) | \(1169975873419524361/108425318400\) | \(6939220377600000000\) | \([2, 2]\) | \(331776\) | \(2.6536\) | |
| 8400.cm3 | 8400cf8 | \([0, 1, 0, -8141008, 11533507988]\) | \(-932348627918877961/358766164249920\) | \(-22961034511994880000000\) | \([4]\) | \(663552\) | \(3.0001\) | |
| 8400.cm4 | 8400cf4 | \([0, 1, 0, -1743008, 869567988]\) | \(9150443179640281/184570312500\) | \(11812500000000000000\) | \([2]\) | \(221184\) | \(2.4508\) | |
| 8400.cm5 | 8400cf3 | \([0, 1, 0, -589008, 132035988]\) | \(353108405631241/86318776320\) | \(5524401684480000000\) | \([2]\) | \(165888\) | \(2.3070\) | |
| 8400.cm6 | 8400cf2 | \([0, 1, 0, -231008, -22512012]\) | \(21302308926361/8930250000\) | \(571536000000000000\) | \([2, 2]\) | \(110592\) | \(2.1043\) | |
| 8400.cm7 | 8400cf1 | \([0, 1, 0, -199008, -34224012]\) | \(13619385906841/6048000\) | \(387072000000000\) | \([2]\) | \(55296\) | \(1.7577\) | \(\Gamma_0(N)\)-optimal |
| 8400.cm8 | 8400cf5 | \([0, 1, 0, 768992, -164512012]\) | \(785793873833639/637994920500\) | \(-40831674912000000000\) | \([4]\) | \(221184\) | \(2.4508\) |
Rank
sage: E.rank()
The elliptic curves in class 8400.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.cm do not have complex multiplication.Modular form 8400.2.a.cm
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.
