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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 38829f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38829.h6 | 38829f1 | \([1, 1, 0, 1811, 7288]\) | \(103823/63\) | \(-398245872087\) | \([2]\) | \(40320\) | \(0.91508\) | \(\Gamma_0(N)\)-optimal |
38829.h5 | 38829f2 | \([1, 1, 0, -7434, 49815]\) | \(7189057/3969\) | \(25089489941481\) | \([2, 2]\) | \(80640\) | \(1.2617\) | |
38829.h3 | 38829f3 | \([1, 1, 0, -72149, -7444182]\) | \(6570725617/45927\) | \(290321240751423\) | \([2]\) | \(161280\) | \(1.6082\) | |
38829.h2 | 38829f4 | \([1, 1, 0, -90639, 10450440]\) | \(13027640977/21609\) | \(136598334125841\) | \([2, 2]\) | \(161280\) | \(1.6082\) | |
38829.h4 | 38829f5 | \([1, 1, 0, -62904, 17001447]\) | \(-4354703137/17294403\) | \(-109324200078714747\) | \([2]\) | \(322560\) | \(1.9548\) | |
38829.h1 | 38829f6 | \([1, 1, 0, -1449654, 671203533]\) | \(53297461115137/147\) | \(929240368203\) | \([2]\) | \(322560\) | \(1.9548\) |
Rank
sage: E.rank()
The elliptic curves in class 38829f have rank \(1\).
Complex multiplication
The elliptic curves in class 38829f do not have complex multiplication.Modular form 38829.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.