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SageMath
sage: E = EllipticCurve("cb1")
sage: E.isogeny_class()
Elliptic curves in class 177450.cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
177450.cb1 | 177450iz8 | [1, 1, 0, -1483957400, -22003535520000] | [2] | 63700992 | |
177450.cb2 | 177450iz6 | [1, 1, 0, -92749400, -343818168000] | [2, 2] | 31850496 | |
177450.cb3 | 177450iz7 | [1, 1, 0, -85989400, -396052688000] | [2] | 63700992 | |
177450.cb4 | 177450iz5 | [1, 1, 0, -18410525, -29878254375] | [2] | 21233664 | |
177450.cb5 | 177450iz3 | [1, 1, 0, -6221400, -4541880000] | [2] | 15925248 | |
177450.cb6 | 177450iz2 | [1, 1, 0, -2440025, 769135125] | [2, 2] | 10616832 | |
177450.cb7 | 177450iz1 | [1, 1, 0, -2102025, 1171693125] | [2] | 5308416 | \(\Gamma_0(N)\)-optimal |
177450.cb8 | 177450iz4 | [1, 1, 0, 8122475, 5659572625] | [2] | 21233664 |
Rank
sage: E.rank()
The elliptic curves in class 177450.cb have rank \(0\).
Complex multiplication
The elliptic curves in class 177450.cb do not have complex multiplication.Modular form 177450.2.a.cb
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.