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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4998.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4998.d1 | 4998g3 | \([1, 1, 0, -36775, 2036917]\) | \(46753267515625/11591221248\) | \(1363695588605952\) | \([2]\) | \(25920\) | \(1.6144\) | |
| 4998.d2 | 4998g1 | \([1, 1, 0, -12520, -544256]\) | \(1845026709625/793152\) | \(93313539648\) | \([2]\) | \(8640\) | \(1.0651\) | \(\Gamma_0(N)\)-optimal |
| 4998.d3 | 4998g2 | \([1, 1, 0, -10560, -717912]\) | \(-1107111813625/1228691592\) | \(-144554337107208\) | \([2]\) | \(17280\) | \(1.4116\) | |
| 4998.d4 | 4998g4 | \([1, 1, 0, 88665, 13050549]\) | \(655215969476375/1001033261568\) | \(-117770562190213632\) | \([2]\) | \(51840\) | \(1.9610\) |
Rank
sage: E.rank()
The elliptic curves in class 4998.d have rank \(1\).
Complex multiplication
The elliptic curves in class 4998.d do not have complex multiplication.Modular form 4998.2.a.d
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.
