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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 4032.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.h1 | 4032h5 | \([0, 0, 0, -451596, 116808176]\) | \(53297461115137/147\) | \(28092137472\) | \([2]\) | \(16384\) | \(1.6632\) | |
| 4032.h2 | 4032h4 | \([0, 0, 0, -28236, 1823600]\) | \(13027640977/21609\) | \(4129544208384\) | \([2, 2]\) | \(8192\) | \(1.3167\) | |
| 4032.h3 | 4032h3 | \([0, 0, 0, -22476, -1289104]\) | \(6570725617/45927\) | \(8776786378752\) | \([2]\) | \(8192\) | \(1.3167\) | |
| 4032.h4 | 4032h6 | \([0, 0, 0, -19596, 2960624]\) | \(-4354703137/17294403\) | \(-3305011881443328\) | \([2]\) | \(16384\) | \(1.6632\) | |
| 4032.h5 | 4032h2 | \([0, 0, 0, -2316, 9200]\) | \(7189057/3969\) | \(758487711744\) | \([2, 2]\) | \(4096\) | \(0.97009\) | |
| 4032.h6 | 4032h1 | \([0, 0, 0, 564, 1136]\) | \(103823/63\) | \(-12039487488\) | \([2]\) | \(2048\) | \(0.62351\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4032.h have rank \(0\).
Complex multiplication
The elliptic curves in class 4032.h do not have complex multiplication.Modular form 4032.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
