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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 609.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 609.a1 | 609b4 | \([1, 1, 1, -204624, 35542050]\) | \(947531277805646290177/38367\) | \(38367\) | \([4]\) | \(1536\) | \(1.2907\) | |
| 609.a2 | 609b5 | \([1, 1, 1, -42469, -2756140]\) | \(8471112631466271697/1662662681263647\) | \(1662662681263647\) | \([2]\) | \(3072\) | \(1.6372\) | |
| 609.a3 | 609b3 | \([1, 1, 1, -13034, 528806]\) | \(244883173420511137/18418027974129\) | \(18418027974129\) | \([2, 2]\) | \(1536\) | \(1.2907\) | |
| 609.a4 | 609b2 | \([1, 1, 1, -12789, 551346]\) | \(231331938231569617/1472026689\) | \(1472026689\) | \([2, 4]\) | \(768\) | \(0.94409\) | |
| 609.a5 | 609b1 | \([1, 1, 1, -784, 8720]\) | \(-53297461115137/4513839183\) | \(-4513839183\) | \([4]\) | \(384\) | \(0.59751\) | \(\Gamma_0(N)\)-optimal |
| 609.a6 | 609b6 | \([1, 1, 1, 12481, 2376092]\) | \(215015459663151503/2552757445339983\) | \(-2552757445339983\) | \([2]\) | \(3072\) | \(1.6372\) |
Rank
sage: E.rank()
The elliptic curves in class 609.a have rank \(1\).
Complex multiplication
The elliptic curves in class 609.a do not have complex multiplication.Modular form 609.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
