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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 51714.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.h1 | 51714f3 | \([1, -1, 0, -1141542, 355203252]\) | \(46753267515625/11591221248\) | \(40786537448770633728\) | \([2]\) | \(1327104\) | \(2.4732\) | |
| 51714.h2 | 51714f1 | \([1, -1, 0, -388647, -93125187]\) | \(1845026709625/793152\) | \(2790898651524672\) | \([2]\) | \(442368\) | \(1.9239\) | \(\Gamma_0(N)\)-optimal |
| 51714.h3 | 51714f2 | \([1, -1, 0, -327807, -123313995]\) | \(-1107111813625/1228691592\) | \(-4323450873543157512\) | \([2]\) | \(884736\) | \(2.2705\) | |
| 51714.h4 | 51714f4 | \([1, -1, 0, 2752218, 2249906868]\) | \(655215969476375/1001033261568\) | \(-3522379543695881077248\) | \([2]\) | \(2654208\) | \(2.8198\) |
Rank
sage: E.rank()
The elliptic curves in class 51714.h have rank \(1\).
Complex multiplication
The elliptic curves in class 51714.h do not have complex multiplication.Modular form 51714.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
