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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 10830p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10830.p4 | 10830p1 | \([1, 0, 1, -685908, -228577454]\) | \(-758575480593601/40535043840\) | \(-1907006848826423040\) | \([2]\) | \(345600\) | \(2.2666\) | \(\Gamma_0(N)\)-optimal |
| 10830.p3 | 10830p2 | \([1, 0, 1, -11111588, -14257372462]\) | \(3225005357698077121/8526675600\) | \(401144965603203600\) | \([2, 2]\) | \(691200\) | \(2.6131\) | |
| 10830.p1 | 10830p3 | \([1, 0, 1, -177785288, -912428607022]\) | \(13209596798923694545921/92340\) | \(4344216651540\) | \([2]\) | \(1382400\) | \(2.9597\) | |
| 10830.p2 | 10830p4 | \([1, 0, 1, -11248768, -13887315694]\) | \(3345930611358906241/165622259047500\) | \(7791845090099858347500\) | \([4]\) | \(1382400\) | \(2.9597\) |
Rank
sage: E.rank()
The elliptic curves in class 10830p have rank \(0\).
Complex multiplication
The elliptic curves in class 10830p do not have complex multiplication.Modular form 10830.2.a.p
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.
