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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 89232bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89232.z5 | 89232bd1 | \([0, -1, 0, -64952, -9118032]\) | \(-1532808577/938223\) | \(-18549240710787072\) | \([2]\) | \(688128\) | \(1.8236\) | \(\Gamma_0(N)\)-optimal |
| 89232.z4 | 89232bd2 | \([0, -1, 0, -1160072, -480457680]\) | \(8732907467857/1656369\) | \(32747424958550016\) | \([2, 2]\) | \(1376256\) | \(2.1701\) | |
| 89232.z3 | 89232bd3 | \([0, -1, 0, -1281752, -373379280]\) | \(11779205551777/3763454409\) | \(74405788108598808576\) | \([2, 2]\) | \(2752512\) | \(2.5167\) | |
| 89232.z1 | 89232bd4 | \([0, -1, 0, -18560312, -30770795472]\) | \(35765103905346817/1287\) | \(25444774637568\) | \([2]\) | \(2752512\) | \(2.5167\) | |
| 89232.z6 | 89232bd5 | \([0, -1, 0, 3626008, -2548498512]\) | \(266679605718863/296110251723\) | \(-5854280204324216451072\) | \([2]\) | \(5505024\) | \(2.8633\) | |
| 89232.z2 | 89232bd6 | \([0, -1, 0, -8136392, 8652810672]\) | \(3013001140430737/108679952667\) | \(2148668922481252773888\) | \([2]\) | \(5505024\) | \(2.8633\) |
Rank
sage: E.rank()
The elliptic curves in class 89232bd have rank \(0\).
Complex multiplication
The elliptic curves in class 89232bd do not have complex multiplication.Modular form 89232.2.a.bd
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
