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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 56784.cq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56784.cq1 | 56784cn4 | \([0, 1, 0, -3634232, -2667868332]\) | \(268498407453697/252\) | \(4982193635328\) | \([2]\) | \(737280\) | \(2.1647\) | |
| 56784.cq2 | 56784cn6 | \([0, 1, 0, -2471512, 1480338068]\) | \(84448510979617/933897762\) | \(18463728118585171968\) | \([2]\) | \(1474560\) | \(2.5113\) | |
| 56784.cq3 | 56784cn3 | \([0, 1, 0, -281272, -20414380]\) | \(124475734657/63011844\) | \(1245782571932860416\) | \([2, 2]\) | \(737280\) | \(2.1647\) | |
| 56784.cq4 | 56784cn2 | \([0, 1, 0, -227192, -41721900]\) | \(65597103937/63504\) | \(1255512796102656\) | \([2, 2]\) | \(368640\) | \(1.8181\) | |
| 56784.cq5 | 56784cn1 | \([0, 1, 0, -10872, -967212]\) | \(-7189057/16128\) | \(-318860392660992\) | \([2]\) | \(184320\) | \(1.4716\) | \(\Gamma_0(N)\)-optimal |
| 56784.cq6 | 56784cn5 | \([0, 1, 0, 1043688, -156620268]\) | \(6359387729183/4218578658\) | \(-83403871984198950912\) | \([2]\) | \(1474560\) | \(2.5113\) |
Rank
sage: E.rank()
The elliptic curves in class 56784.cq have rank \(1\).
Complex multiplication
The elliptic curves in class 56784.cq do not have complex multiplication.Modular form 56784.2.a.cq
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.
