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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 21840bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21840.h4 | 21840bc1 | \([0, -1, 0, 104, -464]\) | \(30080231/36855\) | \(-150958080\) | \([2]\) | \(6144\) | \(0.25550\) | \(\Gamma_0(N)\)-optimal |
21840.h3 | 21840bc2 | \([0, -1, 0, -616, -3920]\) | \(6321363049/1863225\) | \(7631769600\) | \([2, 2]\) | \(12288\) | \(0.60207\) | |
21840.h2 | 21840bc3 | \([0, -1, 0, -3736, 85936]\) | \(1408317602329/58524375\) | \(239715840000\) | \([2]\) | \(24576\) | \(0.94865\) | |
21840.h1 | 21840bc4 | \([0, -1, 0, -9016, -326480]\) | \(19790357598649/2998905\) | \(12283514880\) | \([2]\) | \(24576\) | \(0.94865\) |
Rank
sage: E.rank()
The elliptic curves in class 21840bc have rank \(1\).
Complex multiplication
The elliptic curves in class 21840bc do not have complex multiplication.Modular form 21840.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.