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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 9680bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9680.bb4 | 9680bc1 | \([0, -1, 0, -5485, 154992]\) | \(643956736/15125\) | \(428717762000\) | \([2]\) | \(17280\) | \(1.0175\) | \(\Gamma_0(N)\)-optimal |
| 9680.bb3 | 9680bc2 | \([0, -1, 0, -12140, -286900]\) | \(436334416/171875\) | \(77948684000000\) | \([2]\) | \(34560\) | \(1.3641\) | |
| 9680.bb2 | 9680bc3 | \([0, -1, 0, -53885, -4735828]\) | \(610462990336/8857805\) | \(251074270137680\) | \([2]\) | \(51840\) | \(1.5668\) | |
| 9680.bb1 | 9680bc4 | \([0, -1, 0, -859140, -306223300]\) | \(154639330142416/33275\) | \(15090865222400\) | \([2]\) | \(103680\) | \(1.9134\) |
Rank
sage: E.rank()
The elliptic curves in class 9680bc have rank \(0\).
Complex multiplication
The elliptic curves in class 9680bc do not have complex multiplication.Modular form 9680.2.a.bc
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.
