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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1830.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1830.g1 | 1830g5 | \([1, 1, 1, -81333335, -282359960035]\) | \(59501710967615564842432531441/100055250000\) | \(100055250000\) | \([2]\) | \(92160\) | \(2.7465\) | |
| 1830.g2 | 1830g3 | \([1, 1, 1, -5083335, -4413460035]\) | \(14526798467252802652531441/18837562500000000\) | \(18837562500000000\) | \([2, 2]\) | \(46080\) | \(2.3999\) | |
| 1830.g3 | 1830g6 | \([1, 1, 1, -5039415, -4493412003]\) | \(-14153507863526516575422961/523567199707031250000\) | \(-523567199707031250000\) | \([2]\) | \(92160\) | \(2.7465\) | |
| 1830.g4 | 1830g4 | \([1, 1, 1, -800455, 182367677]\) | \(56719776559071967726321/18403902047738976000\) | \(18403902047738976000\) | \([4]\) | \(46080\) | \(2.3999\) | |
| 1830.g5 | 1830g2 | \([1, 1, 1, -320455, -67808323]\) | \(3639359463108865006321/127603270656000000\) | \(127603270656000000\) | \([2, 4]\) | \(23040\) | \(2.0533\) | |
| 1830.g6 | 1830g1 | \([1, 1, 1, 7225, -3714115]\) | \(41709358422320399/5993089990656000\) | \(-5993089990656000\) | \([4]\) | \(11520\) | \(1.7068\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1830.g have rank \(0\).
Complex multiplication
The elliptic curves in class 1830.g do not have complex multiplication.Modular form 1830.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
