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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 333960co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333960.co4 | 333960co1 | \([0, 1, 0, -478716, 815851584]\) | \(-26752376766544/618796614375\) | \(-280636402933450080000\) | \([2]\) | \(8847360\) | \(2.6044\) | \(\Gamma_0(N)\)-optimal |
| 333960.co3 | 333960co2 | \([0, 1, 0, -16356336, 25343598960]\) | \(266763091319403556/1355769140625\) | \(2459471600163600000000\) | \([2, 2]\) | \(17694720\) | \(2.9509\) | |
| 333960.co1 | 333960co3 | \([0, 1, 0, -261381336, 1626434958960]\) | \(544328872410114151778/14166950625\) | \(51399920058727680000\) | \([2]\) | \(35389440\) | \(3.2975\) | |
| 333960.co2 | 333960co4 | \([0, 1, 0, -25373256, -5717887056]\) | \(497927680189263938/284271240234375\) | \(1031380669687500000000000\) | \([2]\) | \(35389440\) | \(3.2975\) |
Rank
sage: E.rank()
The elliptic curves in class 333960co have rank \(1\).
Complex multiplication
The elliptic curves in class 333960co do not have complex multiplication.Modular form 333960.2.a.co
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.
