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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 93600dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
93600.k3 | 93600dp1 | \([0, 0, 0, -3660825, 1525579000]\) | \(7442744143086784/2927948765625\) | \(2134474650140625000000\) | \([2, 2]\) | \(4718592\) | \(2.7912\) | \(\Gamma_0(N)\)-optimal |
93600.k4 | 93600dp2 | \([0, 0, 0, 11739300, 11012056000]\) | \(3834800837445824/3342041015625\) | \(-155926265625000000000000\) | \([2]\) | \(9437184\) | \(3.1378\) | |
93600.k2 | 93600dp3 | \([0, 0, 0, -26442075, -51258577250]\) | \(350584567631475848/8259273550125\) | \(48168083344329000000000\) | \([2]\) | \(9437184\) | \(3.1378\) | |
93600.k1 | 93600dp4 | \([0, 0, 0, -51192075, 140934735250]\) | \(2543984126301795848/909361981125\) | \(5303399073921000000000\) | \([2]\) | \(9437184\) | \(3.1378\) |
Rank
sage: E.rank()
The elliptic curves in class 93600dp have rank \(1\).
Complex multiplication
The elliptic curves in class 93600dp do not have complex multiplication.Modular form 93600.2.a.dp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.