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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 117117.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117117.bg1 | 117117n1 | \([0, 0, 1, -135876, -19278126]\) | \(-78843215872/539\) | \(-1896602887179\) | \([]\) | \(432000\) | \(1.5368\) | \(\Gamma_0(N)\)-optimal |
| 117117.bg2 | 117117n2 | \([0, 0, 1, -75036, -36579501]\) | \(-13278380032/156590819\) | \(-551002967386130259\) | \([]\) | \(1296000\) | \(2.0861\) | |
| 117117.bg3 | 117117n3 | \([0, 0, 1, 670254, 946830654]\) | \(9463555063808/115539436859\) | \(-406553672597059686699\) | \([]\) | \(3888000\) | \(2.6354\) |
Rank
sage: E.rank()
The elliptic curves in class 117117.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 117117.bg do not have complex multiplication.Modular form 117117.2.a.bg
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
