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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 45084h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45084.f2 | 45084h1 | \([0, -1, 0, -911313, 335149650]\) | \(216727177216000/2738853\) | \(1057748052293712\) | \([2]\) | \(414720\) | \(2.0292\) | \(\Gamma_0(N)\)-optimal |
| 45084.f3 | 45084h2 | \([0, -1, 0, -886748, 354045048]\) | \(-12479332642000/1526829993\) | \(-9434614862670596352\) | \([2]\) | \(829440\) | \(2.3758\) | |
| 45084.f1 | 45084h3 | \([0, -1, 0, -1431513, -88511634]\) | \(840033089536000/477272151837\) | \(184323031947905028048\) | \([2]\) | \(1244160\) | \(2.5785\) | |
| 45084.f4 | 45084h4 | \([0, -1, 0, 5667772, -710409000]\) | \(3258571509326000/1920843121977\) | \(-11869307749093185001728\) | \([2]\) | \(2488320\) | \(2.9251\) |
Rank
sage: E.rank()
The elliptic curves in class 45084h have rank \(1\).
Complex multiplication
The elliptic curves in class 45084h do not have complex multiplication.Modular form 45084.2.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 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.
