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SageMath
E = EllipticCurve("pp1")
E.isogeny_class()
Elliptic curves in class 187200pp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.fg2 | 187200pp1 | \([0, 0, 0, -348300, 88182000]\) | \(-57960603/8125\) | \(-655050240000000000\) | \([2]\) | \(2359296\) | \(2.1500\) | \(\Gamma_0(N)\)-optimal |
| 187200.fg1 | 187200pp2 | \([0, 0, 0, -5748300, 5304582000]\) | \(260549802603/4225\) | \(340626124800000000\) | \([2]\) | \(4718592\) | \(2.4965\) |
Rank
sage: E.rank()
The elliptic curves in class 187200pp have rank \(1\).
Complex multiplication
The elliptic curves in class 187200pp do not have complex multiplication.Modular form 187200.2.a.pp
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.
