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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 7350ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7350.cs7 | 7350ck1 | \([1, 0, 0, -609463, -183114583]\) | \(13619385906841/6048000\) | \(11117830500000000\) | \([2]\) | \(110592\) | \(2.0375\) | \(\Gamma_0(N)\)-optimal |
| 7350.cs6 | 7350ck2 | \([1, 0, 0, -707463, -120296583]\) | \(21302308926361/8930250000\) | \(16416171597656250000\) | \([2, 2]\) | \(221184\) | \(2.3841\) | |
| 7350.cs5 | 7350ck3 | \([1, 0, 0, -1803838, 708532292]\) | \(353108405631241/86318776320\) | \(158676839301120000000\) | \([2]\) | \(331776\) | \(2.5868\) | |
| 7350.cs4 | 7350ck4 | \([1, 0, 0, -5337963, 4663009917]\) | \(9150443179640281/184570312500\) | \(339289260864257812500\) | \([2]\) | \(442368\) | \(2.7306\) | |
| 7350.cs8 | 7350ck5 | \([1, 0, 0, 2355037, -882859083]\) | \(785793873833639/637994920500\) | \(-1172804131279757812500\) | \([2]\) | \(442368\) | \(2.7306\) | |
| 7350.cs2 | 7350ck6 | \([1, 0, 0, -26891838, 53669300292]\) | \(1169975873419524361/108425318400\) | \(199314535694400000000\) | \([2, 2]\) | \(663552\) | \(2.9334\) | |
| 7350.cs1 | 7350ck7 | \([1, 0, 0, -430259838, 3435103244292]\) | \(4791901410190533590281/41160000\) | \(75663013125000000\) | \([2]\) | \(1327104\) | \(3.2799\) | |
| 7350.cs3 | 7350ck8 | \([1, 0, 0, -24931838, 61824860292]\) | \(-932348627918877961/358766164249920\) | \(-659507507153731845000000\) | \([2]\) | \(1327104\) | \(3.2799\) |
Rank
sage: E.rank()
The elliptic curves in class 7350ck have rank \(0\).
Complex multiplication
The elliptic curves in class 7350ck do not have complex multiplication.Modular form 7350.2.a.ck
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
