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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 55770a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55770.d6 | 55770a1 | \([1, 1, 0, 43092, 345168]\) | \(1833318007919/1070530560\) | \(-5167246541783040\) | \([2]\) | \(368640\) | \(1.7045\) | \(\Gamma_0(N)\)-optimal |
55770.d5 | 55770a2 | \([1, 1, 0, -173228, 2551632]\) | \(119102750067601/68309049600\) | \(329714735390726400\) | \([2, 2]\) | \(737280\) | \(2.0511\) | |
55770.d3 | 55770a3 | \([1, 1, 0, -1809148, -933521792]\) | \(135670761487282321/643043610000\) | \(3103848684140490000\) | \([2, 2]\) | \(1474560\) | \(2.3977\) | |
55770.d2 | 55770a4 | \([1, 1, 0, -1998428, 1084165152]\) | \(182864522286982801/463015182960\) | \(2234885852247974640\) | \([2]\) | \(1474560\) | \(2.3977\) | |
55770.d4 | 55770a5 | \([1, 1, 0, -879648, -1890349092]\) | \(-15595206456730321/310672490129100\) | \(-1499556771407551041900\) | \([2]\) | \(2949120\) | \(2.7442\) | |
55770.d1 | 55770a6 | \([1, 1, 0, -28913368, -59852675228]\) | \(553808571467029327441/12529687500\) | \(60478408392187500\) | \([2]\) | \(2949120\) | \(2.7442\) |
Rank
sage: E.rank()
The elliptic curves in class 55770a have rank \(1\).
Complex multiplication
The elliptic curves in class 55770a do not have complex multiplication.Modular form 55770.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.