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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 22050d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.bm3 | 22050d1 | \([1, -1, 0, -57192, -4598784]\) | \(416832723/56000\) | \(2779457625000000\) | \([2]\) | \(110592\) | \(1.6909\) | \(\Gamma_0(N)\)-optimal |
| 22050.bm4 | 22050d2 | \([1, -1, 0, 89808, -24443784]\) | \(1613964717/6125000\) | \(-304003177734375000\) | \([2]\) | \(221184\) | \(2.0375\) | |
| 22050.bm1 | 22050d3 | \([1, -1, 0, -1159692, 480378716]\) | \(4767078987/6860\) | \(248212514556562500\) | \([2]\) | \(331776\) | \(2.2402\) | |
| 22050.bm2 | 22050d4 | \([1, -1, 0, -828942, 759862466]\) | \(-1740992427/5882450\) | \(-212842231232252343750\) | \([2]\) | \(663552\) | \(2.5868\) |
Rank
sage: E.rank()
The elliptic curves in class 22050d have rank \(0\).
Complex multiplication
The elliptic curves in class 22050d do not have complex multiplication.Modular form 22050.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
