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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 163170ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.ds1 | 163170ba1 | \([1, -1, 1, -23603, 1401621]\) | \(-16954786009/370\) | \(-31733464770\) | \([]\) | \(326592\) | \(1.1310\) | \(\Gamma_0(N)\)-optimal |
| 163170.ds2 | 163170ba2 | \([1, -1, 1, -8168, 3185907]\) | \(-702595369/50653000\) | \(-4344311327013000\) | \([]\) | \(979776\) | \(1.6804\) | |
| 163170.ds3 | 163170ba3 | \([1, -1, 1, 73417, -85382769]\) | \(510273943271/37000000000\) | \(-3173346477000000000\) | \([]\) | \(2939328\) | \(2.2297\) |
Rank
sage: E.rank()
The elliptic curves in class 163170ba have rank \(1\).
Complex multiplication
The elliptic curves in class 163170ba do not have complex multiplication.Modular form 163170.2.a.ba
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
