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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 15210.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15210.bc1 | 15210bh2 | \([1, -1, 1, -69998, 6403011]\) | \(10779215329/1232010\) | \(4335127500989610\) | \([2]\) | \(129024\) | \(1.7327\) | |
15210.bc2 | 15210bh1 | \([1, -1, 1, 6052, 501531]\) | \(6967871/35100\) | \(-123507906011100\) | \([2]\) | \(64512\) | \(1.3862\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15210.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 15210.bc do not have complex multiplication.Modular form 15210.2.a.bc
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.