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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 176400dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.p4 | 176400dp1 | \([0, 0, 0, 73500, 2615375]\) | \(2048000/1323\) | \(-28367144520750000\) | \([2]\) | \(1327104\) | \(1.8459\) | \(\Gamma_0(N)\)-optimal |
| 176400.p3 | 176400dp2 | \([0, 0, 0, -312375, 21523250]\) | \(9826000/5103\) | \(1750658061852000000\) | \([2]\) | \(2654208\) | \(2.1925\) | |
| 176400.p2 | 176400dp3 | \([0, 0, 0, -1249500, 553644875]\) | \(-10061824000/352947\) | \(-7567723777146750000\) | \([2]\) | \(3981312\) | \(2.3952\) | |
| 176400.p1 | 176400dp4 | \([0, 0, 0, -20157375, 34833622250]\) | \(2640279346000/3087\) | \(1059040062108000000\) | \([2]\) | \(7962624\) | \(2.7418\) |
Rank
sage: E.rank()
The elliptic curves in class 176400dp have rank \(0\).
Complex multiplication
The elliptic curves in class 176400dp do not have complex multiplication.Modular form 176400.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
