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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2046.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2046.f1 | 2046g4 | \([1, 1, 1, -349184, 79274465]\) | \(4708545773991716929537/65472\) | \(65472\) | \([4]\) | \(6144\) | \(1.4103\) | |
| 2046.f2 | 2046g5 | \([1, 1, 1, -65704, -5009503]\) | \(31368919137792368257/7430386718185992\) | \(7430386718185992\) | \([2]\) | \(12288\) | \(1.7569\) | |
| 2046.f3 | 2046g3 | \([1, 1, 1, -22144, 1193441]\) | \(1200862149227882497/70094268661824\) | \(70094268661824\) | \([2, 2]\) | \(6144\) | \(1.4103\) | |
| 2046.f4 | 2046g2 | \([1, 1, 1, -21824, 1231841]\) | \(1149550394446181377/4286582784\) | \(4286582784\) | \([2, 4]\) | \(3072\) | \(1.0637\) | |
| 2046.f5 | 2046g1 | \([1, 1, 1, -1344, 19425]\) | \(-268498407453697/17163091968\) | \(-17163091968\) | \([4]\) | \(1536\) | \(0.71716\) | \(\Gamma_0(N)\)-optimal |
| 2046.f6 | 2046g6 | \([1, 1, 1, 16296, 4945185]\) | \(478591624936623743/10812469457036808\) | \(-10812469457036808\) | \([2]\) | \(12288\) | \(1.7569\) |
Rank
sage: E.rank()
The elliptic curves in class 2046.f have rank \(0\).
Complex multiplication
The elliptic curves in class 2046.f do not have complex multiplication.Modular form 2046.2.a.f
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.
