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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 259920ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259920.ew3 | 259920ew1 | \([0, 0, 0, -1344879147, -19157631147686]\) | \(-1914980734749238129/20440940544000\) | \(-2871507513701340624715776000\) | \([2]\) | \(199065600\) | \(4.0852\) | \(\Gamma_0(N)\)-optimal |
| 259920.ew2 | 259920ew2 | \([0, 0, 0, -21572893227, -1219581035519654]\) | \(7903870428425797297009/886464000000\) | \(124528909574707347456000000\) | \([2]\) | \(398131200\) | \(4.4317\) | |
| 259920.ew4 | 259920ew3 | \([0, 0, 0, 4444059093, -99724041277094]\) | \(69096190760262356111/70568821500000000\) | \(-9913384402939277531136000000000\) | \([2]\) | \(597196800\) | \(4.6345\) | |
| 259920.ew1 | 259920ew4 | \([0, 0, 0, -24080601387, -918421731577766]\) | \(10993009831928446009969/3767761230468750000\) | \(529288496282004750000000000000000\) | \([2]\) | \(1194393600\) | \(4.9810\) |
Rank
sage: E.rank()
The elliptic curves in class 259920ew have rank \(1\).
Complex multiplication
The elliptic curves in class 259920ew do not have complex multiplication.Modular form 259920.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
