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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5200.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5200.e1 | 5200s3 | \([0, 1, 0, -83008, 8807988]\) | \(988345570681/44994560\) | \(2879651840000000\) | \([2]\) | \(41472\) | \(1.7283\) | |
| 5200.e2 | 5200s1 | \([0, 1, 0, -13008, -572012]\) | \(3803721481/26000\) | \(1664000000000\) | \([2]\) | \(13824\) | \(1.1790\) | \(\Gamma_0(N)\)-optimal |
| 5200.e3 | 5200s2 | \([0, 1, 0, -5008, -1260012]\) | \(-217081801/10562500\) | \(-676000000000000\) | \([2]\) | \(27648\) | \(1.5256\) | |
| 5200.e4 | 5200s4 | \([0, 1, 0, 44992, 33639988]\) | \(157376536199/7722894400\) | \(-494265241600000000\) | \([2]\) | \(82944\) | \(2.0749\) |
Rank
sage: E.rank()
The elliptic curves in class 5200.e have rank \(0\).
Complex multiplication
The elliptic curves in class 5200.e do not have complex multiplication.Modular form 5200.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
