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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 229320.ds
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
229320.ds1 | 229320o4 | \([0, 0, 0, -247665747, -1500192649186]\) | \(19129597231400697604/26325\) | \(2311980170572800\) | \([2]\) | \(14155776\) | \(3.1144\) | |
229320.ds2 | 229320o2 | \([0, 0, 0, -15479247, -23440071886]\) | \(18681746265374416/693005625\) | \(15215719497582240000\) | \([2, 2]\) | \(7077888\) | \(2.7678\) | |
229320.ds3 | 229320o3 | \([0, 0, 0, -14764827, -25701496954]\) | \(-4053153720264484/903687890625\) | \(-79365944292944400000000\) | \([2]\) | \(14155776\) | \(3.1144\) | |
229320.ds4 | 229320o1 | \([0, 0, 0, -1012242, -330478099]\) | \(83587439220736/13990184325\) | \(19198141466084053200\) | \([2]\) | \(3538944\) | \(2.4212\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 229320.ds have rank \(1\).
Complex multiplication
The elliptic curves in class 229320.ds do not have complex multiplication.Modular form 229320.2.a.ds
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.