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SageMath
E = EllipticCurve("qr1")
E.isogeny_class()
Elliptic curves in class 235200qr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.qr7 | 235200qr1 | \([0, 1, 0, 16462367, 19378152863]\) | \(1023887723039/928972800\) | \(-447662984999731200000000\) | \([2]\) | \(28311552\) | \(3.2255\) | \(\Gamma_0(N)\)-optimal |
| 235200.qr6 | 235200qr2 | \([0, 1, 0, -83889633, 173418472863]\) | \(135487869158881/51438240000\) | \(24787589110824960000000000\) | \([2, 2]\) | \(56623104\) | \(3.5721\) | |
| 235200.qr4 | 235200qr3 | \([0, 1, 0, -1181489633, 15626528872863]\) | \(378499465220294881/120530818800\) | \(58082632912900915200000000\) | \([2]\) | \(113246208\) | \(3.9187\) | |
| 235200.qr5 | 235200qr4 | \([0, 1, 0, -591921633, -5419505815137]\) | \(47595748626367201/1215506250000\) | \(585740676326400000000000000\) | \([2, 2]\) | \(113246208\) | \(3.9187\) | |
| 235200.qr8 | 235200qr5 | \([0, 1, 0, 99566367, -17323471735137]\) | \(226523624554079/269165039062500\) | \(-129708022500000000000000000000\) | \([2]\) | \(226492416\) | \(4.2653\) | |
| 235200.qr2 | 235200qr6 | \([0, 1, 0, -9411921633, -351454565815137]\) | \(191342053882402567201/129708022500\) | \(62505038393763840000000000\) | \([2, 2]\) | \(226492416\) | \(4.2653\) | |
| 235200.qr3 | 235200qr7 | \([0, 1, 0, -9353121633, -356062545415137]\) | \(-187778242790732059201/4984939585440150\) | \(-2402194052249387857305600000000\) | \([2]\) | \(452984832\) | \(4.6118\) | |
| 235200.qr1 | 235200qr8 | \([0, 1, 0, -150590721633, -22492949306215137]\) | \(783736670177727068275201/360150\) | \(173552792985600000000\) | \([2]\) | \(452984832\) | \(4.6118\) |
Rank
sage: E.rank()
The elliptic curves in class 235200qr have rank \(0\).
Complex multiplication
The elliptic curves in class 235200qr do not have complex multiplication.Modular form 235200.2.a.qr
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
