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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 302016dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
302016.dp3 | 302016dp1 | \([0, -1, 0, -19037, 1017117]\) | \(420616192/117\) | \(212247180288\) | \([2]\) | \(737280\) | \(1.1552\) | \(\Gamma_0(N)\)-optimal |
302016.dp2 | 302016dp2 | \([0, -1, 0, -21457, 744625]\) | \(37642192/13689\) | \(397326721499136\) | \([2, 2]\) | \(1474560\) | \(1.5018\) | |
302016.dp4 | 302016dp3 | \([0, -1, 0, 65663, 5187745]\) | \(269676572/257049\) | \(-29843651525935104\) | \([2]\) | \(2949120\) | \(1.8484\) | |
302016.dp1 | 302016dp4 | \([0, -1, 0, -147297, -21176703]\) | \(3044193988/85293\) | \(9902604443516928\) | \([2]\) | \(2949120\) | \(1.8484\) |
Rank
sage: E.rank()
The elliptic curves in class 302016dp have rank \(0\).
Complex multiplication
The elliptic curves in class 302016dp do not have complex multiplication.Modular form 302016.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.