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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 235200ew
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.ew1 | 235200ew1 | \([0, -1, 0, -10208, 378162]\) | \(1000000/63\) | \(7411887000000\) | \([2]\) | \(442368\) | \(1.2206\) | \(\Gamma_0(N)\)-optimal |
235200.ew2 | 235200ew2 | \([0, -1, 0, 8167, 1572537]\) | \(8000/147\) | \(-1106841792000000\) | \([2]\) | \(884736\) | \(1.5671\) |
Rank
sage: E.rank()
The elliptic curves in class 235200ew have rank \(0\).
Complex multiplication
The elliptic curves in class 235200ew do not have complex multiplication.Modular form 235200.2.a.ew
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.