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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 75810co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75810.cn7 | 75810co1 | \([1, 1, 1, -179605, -29360773]\) | \(13619385906841/6048000\) | \(284533488288000\) | \([2]\) | \(580608\) | \(1.7320\) | \(\Gamma_0(N)\)-optimal |
75810.cn6 | 75810co2 | \([1, 1, 1, -208485, -19322085]\) | \(21302308926361/8930250000\) | \(420131478800250000\) | \([2, 2]\) | \(1161216\) | \(2.0786\) | |
75810.cn5 | 75810co3 | \([1, 1, 1, -531580, 113151197]\) | \(353108405631241/86318776320\) | \(4060942878816337920\) | \([2]\) | \(1741824\) | \(2.2813\) | |
75810.cn8 | 75810co4 | \([1, 1, 1, 694015, -140979085]\) | \(785793873833639/637994920500\) | \(-30015033108447460500\) | \([2]\) | \(2322432\) | \(2.4252\) | |
75810.cn4 | 75810co5 | \([1, 1, 1, -1573065, 745388547]\) | \(9150443179640281/184570312500\) | \(8683272958007812500\) | \([2]\) | \(2322432\) | \(2.4252\) | |
75810.cn2 | 75810co6 | \([1, 1, 1, -7924860, 8582892765]\) | \(1169975873419524361/108425318400\) | \(5100964626833510400\) | \([2, 2]\) | \(3483648\) | \(2.6279\) | |
75810.cn3 | 75810co7 | \([1, 1, 1, -7347260, 9887806685]\) | \(-932348627918877961/358766164249920\) | \(-16878470270128190579520\) | \([2]\) | \(6967296\) | \(2.9745\) | |
75810.cn1 | 75810co8 | \([1, 1, 1, -126794940, 549489304797]\) | \(4791901410190533590281/41160000\) | \(1936408461960000\) | \([2]\) | \(6967296\) | \(2.9745\) |
Rank
sage: E.rank()
The elliptic curves in class 75810co have rank \(1\).
Complex multiplication
The elliptic curves in class 75810co do not have complex multiplication.Modular form 75810.2.a.co
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.