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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 193200.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.cr1 | 193200dx4 | \([0, -1, 0, -14768040008, -690738794857488]\) | \(5565604209893236690185614401/229307220930246900000\) | \(14675662139535801600000000000\) | \([2]\) | \(353894400\) | \(4.4860\) | |
| 193200.cr2 | 193200dx3 | \([0, -1, 0, -4502312008, 107232522902512]\) | \(157706830105239346386477121/13650704956054687500000\) | \(873645117187500000000000000000\) | \([2]\) | \(353894400\) | \(4.4860\) | |
| 193200.cr3 | 193200dx2 | \([0, -1, 0, -968040008, -9681194857488]\) | \(1567558142704512417614401/274462175610000000000\) | \(17565579239040000000000000000\) | \([2, 2]\) | \(176947200\) | \(4.1395\) | |
| 193200.cr4 | 193200dx1 | \([0, -1, 0, 115351992, -866717545488]\) | \(2652277923951208297919/6605028468326400000\) | \(-422721821972889600000000000\) | \([2]\) | \(88473600\) | \(3.7929\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 193200.cr do not have complex multiplication.Modular form 193200.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
