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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 443760bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
443760.bf4 | 443760bf1 | \([0, -1, 0, 43760, 32100352]\) | \(357911/17415\) | \(-450914457593180160\) | \([2]\) | \(5203968\) | \(2.0672\) | \(\Gamma_0(N)\)-optimal* |
443760.bf3 | 443760bf2 | \([0, -1, 0, -1287520, 540116800]\) | \(9116230969/416025\) | \(10771845375837081600\) | \([2, 2]\) | \(10407936\) | \(2.4138\) | \(\Gamma_0(N)\)-optimal* |
443760.bf1 | 443760bf3 | \([0, -1, 0, -20369200, 35390897152]\) | \(36097320816649/80625\) | \(2087566933301760000\) | \([4]\) | \(20815872\) | \(2.7604\) | \(\Gamma_0(N)\)-optimal* |
443760.bf2 | 443760bf4 | \([0, -1, 0, -3506320, -1818911360]\) | \(184122897769/51282015\) | \(1327809473328184258560\) | \([2]\) | \(20815872\) | \(2.7604\) |
Rank
sage: E.rank()
The elliptic curves in class 443760bf have rank \(0\).
Complex multiplication
The elliptic curves in class 443760bf do not have complex multiplication.Modular form 443760.2.a.bf
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.