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SageMath
E = EllipticCurve("lj1")
E.isogeny_class()
Elliptic curves in class 221760.lj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.lj1 | 221760ms4 | \([0, 0, 0, -4472172, -3628509264]\) | \(1917114236485083/7117764500\) | \(36726101417263104000\) | \([2]\) | \(7962624\) | \(2.6141\) | |
221760.lj2 | 221760ms2 | \([0, 0, 0, -290412, 56333744]\) | \(382704614800227/27778076480\) | \(196610114180874240\) | \([2]\) | \(2654208\) | \(2.0648\) | |
221760.lj3 | 221760ms3 | \([0, 0, 0, -152172, -108573264]\) | \(-75526045083/943250000\) | \(-4866962817024000000\) | \([2]\) | \(3981312\) | \(2.2675\) | |
221760.lj4 | 221760ms1 | \([0, 0, 0, 16788, 3863984]\) | \(73929353373/954060800\) | \(-6752735487590400\) | \([2]\) | \(1327104\) | \(1.7182\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.lj have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.lj do not have complex multiplication.Modular form 221760.2.a.lj
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.