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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 6270c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6270.c3 | 6270c1 | \([1, 1, 0, -72, -144]\) | \(42180533641/18810000\) | \(18810000\) | \([2]\) | \(1536\) | \(0.091337\) | \(\Gamma_0(N)\)-optimal |
6270.c2 | 6270c2 | \([1, 1, 0, -572, 4956]\) | \(20753798525641/353816100\) | \(353816100\) | \([2, 2]\) | \(3072\) | \(0.43791\) | |
6270.c1 | 6270c3 | \([1, 1, 0, -9122, 331566]\) | \(83959202297868841/25036110\) | \(25036110\) | \([2]\) | \(6144\) | \(0.78448\) | |
6270.c4 | 6270c4 | \([1, 1, 0, -22, 14746]\) | \(-1263214441/94053968910\) | \(-94053968910\) | \([2]\) | \(6144\) | \(0.78448\) |
Rank
sage: E.rank()
The elliptic curves in class 6270c have rank \(1\).
Complex multiplication
The elliptic curves in class 6270c do not have complex multiplication.Modular form 6270.2.a.c
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.