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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 80850i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.m3 | 80850i1 | \([1, 1, 0, -613750, 160442500]\) | \(13908844989649/1980372240\) | \(3640450213496250000\) | \([2]\) | \(1769472\) | \(2.2873\) | \(\Gamma_0(N)\)-optimal |
| 80850.m2 | 80850i2 | \([1, 1, 0, -2598250, -1452956000]\) | \(1055257664218129/115307784900\) | \(211966337276564062500\) | \([2, 2]\) | \(3538944\) | \(2.6339\) | |
| 80850.m4 | 80850i3 | \([1, 1, 0, 3465500, -7219582250]\) | \(2503876820718671/13702874328990\) | \(-25189522842677257968750\) | \([2]\) | \(7077888\) | \(2.9805\) | |
| 80850.m1 | 80850i4 | \([1, 1, 0, -40414000, -98904143750]\) | \(3971101377248209009/56495958750\) | \(103854578921542968750\) | \([2]\) | \(7077888\) | \(2.9805\) |
Rank
sage: E.rank()
The elliptic curves in class 80850i have rank \(0\).
Complex multiplication
The elliptic curves in class 80850i do not have complex multiplication.Modular form 80850.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
