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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 159120cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159120.r3 | 159120cs1 | \([0, 0, 0, -10968, -444433]\) | \(-337770946363392/2051393825\) | \(-886202132400\) | \([2]\) | \(221184\) | \(1.1318\) | \(\Gamma_0(N)\)-optimal |
159120.r2 | 159120cs2 | \([0, 0, 0, -175743, -28357318]\) | \(86846853774358512/3174665\) | \(21943284480\) | \([2]\) | \(442368\) | \(1.4784\) | |
159120.r4 | 159120cs3 | \([0, 0, 0, 29592, -2370357]\) | \(9099874271232/12973390625\) | \(-4085683962750000\) | \([2]\) | \(663552\) | \(1.6811\) | |
159120.r1 | 159120cs4 | \([0, 0, 0, -189783, -23561982]\) | \(150025256088048/39223549625\) | \(197641504580832000\) | \([2]\) | \(1327104\) | \(2.0277\) |
Rank
sage: E.rank()
The elliptic curves in class 159120cs have rank \(1\).
Complex multiplication
The elliptic curves in class 159120cs do not have complex multiplication.Modular form 159120.2.a.cs
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.