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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 182070do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.r4 | 182070do1 | \([1, -1, 0, -28665, 20976925]\) | \(-148035889/10710000\) | \(-188456242348710000\) | \([2]\) | \(2359296\) | \(1.9945\) | \(\Gamma_0(N)\)-optimal |
182070.r3 | 182070do2 | \([1, -1, 0, -1329165, 586174225]\) | \(14758408587889/114704100\) | \(2018366355554684100\) | \([2, 2]\) | \(4718592\) | \(2.3411\) | |
182070.r1 | 182070do3 | \([1, -1, 0, -21226815, 37647537115]\) | \(60111445514713489/3673530\) | \(64640491125607530\) | \([2]\) | \(9437184\) | \(2.6877\) | |
182070.r2 | 182070do4 | \([1, -1, 0, -2239515, -318167465]\) | \(70593496254289/38358689670\) | \(674970543102565715670\) | \([2]\) | \(9437184\) | \(2.6877\) |
Rank
sage: E.rank()
The elliptic curves in class 182070do have rank \(0\).
Complex multiplication
The elliptic curves in class 182070do do not have complex multiplication.Modular form 182070.2.a.do
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.