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SageMath
E = EllipticCurve("qp1")
E.isogeny_class()
Elliptic curves in class 235200qp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.qp4 | 235200qp1 | \([0, 1, 0, 53492, 2615738]\) | \(143877824/108045\) | \(-12711386205000000\) | \([2]\) | \(1179648\) | \(1.7783\) | \(\Gamma_0(N)\)-optimal |
235200.qp3 | 235200qp2 | \([0, 1, 0, -246633, 22123863]\) | \(220348864/99225\) | \(747118209600000000\) | \([2, 2]\) | \(2359296\) | \(2.1249\) | |
235200.qp1 | 235200qp3 | \([0, 1, 0, -3333633, 2340460863]\) | \(68017239368/39375\) | \(2371803840000000000\) | \([2]\) | \(4718592\) | \(2.4714\) | |
235200.qp2 | 235200qp4 | \([0, 1, 0, -1961633, -1042891137]\) | \(13858588808/229635\) | \(13832359994880000000\) | \([2]\) | \(4718592\) | \(2.4714\) |
Rank
sage: E.rank()
The elliptic curves in class 235200qp have rank \(1\).
Complex multiplication
The elliptic curves in class 235200qp do not have complex multiplication.Modular form 235200.2.a.qp
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.