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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 277200et
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.et4 | 277200et1 | \([0, 0, 0, -2320275, -450040750]\) | \(29609739866953/15259926528\) | \(711967132090368000000\) | \([2]\) | \(11796480\) | \(2.6933\) | \(\Gamma_0(N)\)-optimal |
277200.et2 | 277200et2 | \([0, 0, 0, -20752275, 36063751250]\) | \(21184262604460873/216872764416\) | \(10118415696592896000000\) | \([2, 2]\) | \(23592960\) | \(3.0399\) | |
277200.et1 | 277200et3 | \([0, 0, 0, -331216275, 2320147399250]\) | \(86129359107301290313/9166294368\) | \(427662630033408000000\) | \([2]\) | \(47185920\) | \(3.3865\) | |
277200.et3 | 277200et4 | \([0, 0, 0, -5200275, 88862791250]\) | \(-333345918055753/72923718045024\) | \(-3402328989108639744000000\) | \([2]\) | \(47185920\) | \(3.3865\) |
Rank
sage: E.rank()
The elliptic curves in class 277200et have rank \(0\).
Complex multiplication
The elliptic curves in class 277200et do not have complex multiplication.Modular form 277200.2.a.et
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.