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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 214200ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 214200.bs3 | 214200ew1 | \([0, 0, 0, -26850, -1693375]\) | \(11745974272/357\) | \(65063250000\) | \([2]\) | \(327680\) | \(1.1728\) | \(\Gamma_0(N)\)-optimal |
| 214200.bs2 | 214200ew2 | \([0, 0, 0, -27975, -1543750]\) | \(830321872/127449\) | \(371641284000000\) | \([2, 2]\) | \(655360\) | \(1.5193\) | |
| 214200.bs1 | 214200ew3 | \([0, 0, 0, -122475, 14993750]\) | \(17418812548/1753941\) | \(20457967824000000\) | \([2]\) | \(1310720\) | \(1.8659\) | |
| 214200.bs4 | 214200ew4 | \([0, 0, 0, 48525, -8505250]\) | \(1083360092/3306177\) | \(-38563248528000000\) | \([2]\) | \(1310720\) | \(1.8659\) |
Rank
sage: E.rank()
The elliptic curves in class 214200ew have rank \(1\).
Complex multiplication
The elliptic curves in class 214200ew do not have complex multiplication.Modular form 214200.2.a.ew
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.
