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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 77658.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
77658.t1 | 77658t4 | \([1, 0, 1, -2485095, -1508072162]\) | \(268498407453697/252\) | \(1592983488348\) | \([2]\) | \(1290240\) | \(2.0697\) | |
77658.t2 | 77658t6 | \([1, 0, 1, -1690025, 837384338]\) | \(84448510979617/933897762\) | \(5903506804250596338\) | \([2]\) | \(2580480\) | \(2.4163\) | |
77658.t3 | 77658t3 | \([1, 0, 1, -192335, -11506354]\) | \(124475734657/63011844\) | \(398320742310952356\) | \([2, 2]\) | \(1290240\) | \(2.0697\) | |
77658.t4 | 77658t2 | \([1, 0, 1, -155355, -23561834]\) | \(65597103937/63504\) | \(401431839063696\) | \([2, 2]\) | \(645120\) | \(1.7231\) | |
77658.t5 | 77658t1 | \([1, 0, 1, -7435, -545482]\) | \(-7189057/16128\) | \(-101950943254272\) | \([2]\) | \(322560\) | \(1.3766\) | \(\Gamma_0(N)\)-optimal |
77658.t6 | 77658t5 | \([1, 0, 1, 713675, -88698406]\) | \(6359387729183/4218578658\) | \(-26667167247981208242\) | \([2]\) | \(2580480\) | \(2.4163\) |
Rank
sage: E.rank()
The elliptic curves in class 77658.t have rank \(0\).
Complex multiplication
The elliptic curves in class 77658.t do not have complex multiplication.Modular form 77658.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.