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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 101150o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
101150.w3 | 101150o1 | \([1, -1, 0, -135017, 18822141]\) | \(721734273/13328\) | \(5026648744250000\) | \([2]\) | \(589824\) | \(1.8073\) | \(\Gamma_0(N)\)-optimal |
101150.w2 | 101150o2 | \([1, -1, 0, -279517, -28429359]\) | \(6403769793/2775556\) | \(1046799600990062500\) | \([2, 2]\) | \(1179648\) | \(2.1539\) | |
101150.w4 | 101150o3 | \([1, -1, 0, 948733, -211438609]\) | \(250404380127/196003234\) | \(-73922524764033531250\) | \([2]\) | \(2359296\) | \(2.5005\) | |
101150.w1 | 101150o4 | \([1, -1, 0, -3819767, -2871250109]\) | \(16342588257633/8185058\) | \(3086990660062531250\) | \([2]\) | \(2359296\) | \(2.5005\) |
Rank
sage: E.rank()
The elliptic curves in class 101150o have rank \(0\).
Complex multiplication
The elliptic curves in class 101150o do not have complex multiplication.Modular form 101150.2.a.o
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.