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SageMath
E = EllipticCurve("hj1")
E.isogeny_class()
Elliptic curves in class 221760hj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.l3 | 221760hj1 | \([0, 0, 0, -16908, -4021232]\) | \(-75526045083/943250000\) | \(-6676217856000000\) | \([2]\) | \(1327104\) | \(1.7182\) | \(\Gamma_0(N)\)-optimal |
221760.l2 | 221760hj2 | \([0, 0, 0, -496908, -134389232]\) | \(1917114236485083/7117764500\) | \(50378739941376000\) | \([2]\) | \(2654208\) | \(2.0648\) | |
221760.l4 | 221760hj3 | \([0, 0, 0, 151092, 104327568]\) | \(73929353373/954060800\) | \(-4922744170453401600\) | \([2]\) | \(3981312\) | \(2.2675\) | |
221760.l1 | 221760hj4 | \([0, 0, 0, -2613708, 1521011088]\) | \(382704614800227/27778076480\) | \(143328773237857320960\) | \([2]\) | \(7962624\) | \(2.6141\) |
Rank
sage: E.rank()
The elliptic curves in class 221760hj have rank \(0\).
Complex multiplication
The elliptic curves in class 221760hj do not have complex multiplication.Modular form 221760.2.a.hj
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.