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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 43681.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
43681.g1 | 43681h2 | \([0, 0, 1, -1659878, 823921942]\) | \(-884736\) | \(-571660940565063019\) | \([]\) | \(452200\) | \(2.3210\) | \(-19\) | |
43681.g2 | 43681h1 | \([0, 0, 1, -4598, -120123]\) | \(-884736\) | \(-12151136899\) | \([]\) | \(23800\) | \(0.84880\) | \(\Gamma_0(N)\)-optimal | \(-19\) |
Rank
sage: E.rank()
The elliptic curves in class 43681.g have rank \(0\).
Complex multiplication
Each elliptic curve in class 43681.g has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-19}) \).Modular form 43681.2.a.g
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.
\(\left(\begin{array}{rr} 1 & 19 \\ 19 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.