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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 27600s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27600.cr4 | 27600s1 | \([0, 1, 0, -98908, 76602188]\) | \(-26752376766544/618796614375\) | \(-2475186457500000000\) | \([2]\) | \(294912\) | \(2.2101\) | \(\Gamma_0(N)\)-optimal |
| 27600.cr3 | 27600s2 | \([0, 1, 0, -3379408, 2379513188]\) | \(266763091319403556/1355769140625\) | \(21692306250000000000\) | \([2, 2]\) | \(589824\) | \(2.5567\) | |
| 27600.cr2 | 27600s3 | \([0, 1, 0, -5242408, -537944812]\) | \(497927680189263938/284271240234375\) | \(9096679687500000000000\) | \([2]\) | \(1179648\) | \(2.9033\) | |
| 27600.cr1 | 27600s4 | \([0, 1, 0, -54004408, 152735763188]\) | \(544328872410114151778/14166950625\) | \(453342420000000000\) | \([2]\) | \(1179648\) | \(2.9033\) |
Rank
sage: E.rank()
The elliptic curves in class 27600s have rank \(0\).
Complex multiplication
The elliptic curves in class 27600s do not have complex multiplication.Modular form 27600.2.a.s
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.
