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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 47040.ha
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.ha1 | 47040dm8 | \([0, 1, 0, -6023628865, 179941184998175]\) | \(783736670177727068275201/360150\) | \(11107378751078400\) | \([2]\) | \(18874368\) | \(3.8071\) | |
| 47040.ha2 | 47040dm6 | \([0, 1, 0, -376476865, 2811485935775]\) | \(191342053882402567201/129708022500\) | \(4000322457200885760000\) | \([2, 2]\) | \(9437184\) | \(3.4605\) | |
| 47040.ha3 | 47040dm7 | \([0, 1, 0, -374124865, 2848350713375]\) | \(-187778242790732059201/4984939585440150\) | \(-153740419343960822867558400\) | \([2]\) | \(18874368\) | \(3.8071\) | |
| 47040.ha4 | 47040dm4 | \([0, 1, 0, -47259585, -125031134817]\) | \(378499465220294881/120530818800\) | \(3717288506425658572800\) | \([2]\) | \(4718592\) | \(3.1140\) | |
| 47040.ha5 | 47040dm3 | \([0, 1, 0, -23676865, 43346575775]\) | \(47595748626367201/1215506250000\) | \(37487403284889600000000\) | \([2, 2]\) | \(4718592\) | \(3.1140\) | |
| 47040.ha6 | 47040dm2 | \([0, 1, 0, -3355585, -1388690017]\) | \(135487869158881/51438240000\) | \(1586405703092797440000\) | \([2, 2]\) | \(2359296\) | \(2.7674\) | |
| 47040.ha7 | 47040dm1 | \([0, 1, 0, 658495, -154761825]\) | \(1023887723039/928972800\) | \(-28650431039982796800\) | \([2]\) | \(1179648\) | \(2.4208\) | \(\Gamma_0(N)\)-optimal |
| 47040.ha8 | 47040dm5 | \([0, 1, 0, 3982655, 138589366943]\) | \(226523624554079/269165039062500\) | \(-8301313440000000000000000\) | \([2]\) | \(9437184\) | \(3.4605\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.ha have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.ha do not have complex multiplication.Modular form 47040.2.a.ha
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
