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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4160b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4160.o2 | 4160b1 | \([0, -1, 0, -2081, 37025]\) | \(3803721481/26000\) | \(6815744000\) | \([2]\) | \(4608\) | \(0.72086\) | \(\Gamma_0(N)\)-optimal |
| 4160.o3 | 4160b2 | \([0, -1, 0, -801, 80801]\) | \(-217081801/10562500\) | \(-2768896000000\) | \([2]\) | \(9216\) | \(1.0674\) | |
| 4160.o1 | 4160b3 | \([0, -1, 0, -13281, -561055]\) | \(988345570681/44994560\) | \(11795053936640\) | \([2]\) | \(13824\) | \(1.2702\) | |
| 4160.o4 | 4160b4 | \([0, -1, 0, 7199, -2154399]\) | \(157376536199/7722894400\) | \(-2024510429593600\) | \([2]\) | \(27648\) | \(1.6167\) |
Rank
sage: E.rank()
The elliptic curves in class 4160b have rank \(1\).
Complex multiplication
The elliptic curves in class 4160b do not have complex multiplication.Modular form 4160.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
