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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 215600.he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 215600.he1 | 215600er4 | \([0, -1, 0, -8697908, -9870581188]\) | \(154639330142416/33275\) | \(15659081900000000\) | \([2]\) | \(5971968\) | \(2.4921\) | |
| 215600.he2 | 215600er3 | \([0, -1, 0, -545533, -152950188]\) | \(610462990336/8857805\) | \(260527975111250000\) | \([2]\) | \(2985984\) | \(2.1455\) | |
| 215600.he3 | 215600er2 | \([0, -1, 0, -122908, -9331188]\) | \(436334416/171875\) | \(80883687500000000\) | \([2]\) | \(1990656\) | \(1.9428\) | |
| 215600.he4 | 215600er1 | \([0, -1, 0, -55533, 4952312]\) | \(643956736/15125\) | \(444860281250000\) | \([2]\) | \(995328\) | \(1.5962\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 215600.he have rank \(0\).
Complex multiplication
The elliptic curves in class 215600.he do not have complex multiplication.Modular form 215600.2.a.he
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
