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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 76230du
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.dr7 | 76230du1 | \([1, -1, 1, 228667, 31657277]\) | \(1023887723039/928972800\) | \(-1199738615272243200\) | \([2]\) | \(1310720\) | \(2.1564\) | \(\Gamma_0(N)\)-optimal |
| 76230.dr6 | 76230du2 | \([1, -1, 1, -1165253, 284235581]\) | \(135487869158881/51438240000\) | \(66430839341734560000\) | \([2, 2]\) | \(2621440\) | \(2.5030\) | |
| 76230.dr5 | 76230du3 | \([1, -1, 1, -8221973, -8869741603]\) | \(47595748626367201/1215506250000\) | \(1569787387994306250000\) | \([2, 2]\) | \(5242880\) | \(2.8495\) | |
| 76230.dr4 | 76230du4 | \([1, -1, 1, -16411253, 25586497181]\) | \(378499465220294881/120530818800\) | \(155661691757543017200\) | \([2]\) | \(5242880\) | \(2.8495\) | |
| 76230.dr8 | 76230du5 | \([1, -1, 1, 1383007, -28360167019]\) | \(226523624554079/269165039062500\) | \(-347618026323852539062500\) | \([2]\) | \(10485760\) | \(3.1961\) | |
| 76230.dr2 | 76230du6 | \([1, -1, 1, -130734473, -575318536603]\) | \(191342053882402567201/129708022500\) | \(167513756381081302500\) | \([2, 2]\) | \(10485760\) | \(3.1961\) | |
| 76230.dr3 | 76230du7 | \([1, -1, 1, -129917723, -582862366303]\) | \(-187778242790732059201/4984939585440150\) | \(-6437889801996092491555350\) | \([2]\) | \(20971520\) | \(3.5427\) | |
| 76230.dr1 | 76230du8 | \([1, -1, 1, -2091751223, -36821967736903]\) | \(783736670177727068275201/360150\) | \(465122189035350\) | \([2]\) | \(20971520\) | \(3.5427\) |
Rank
sage: E.rank()
The elliptic curves in class 76230du have rank \(1\).
Complex multiplication
The elliptic curves in class 76230du do not have complex multiplication.Modular form 76230.2.a.du
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
