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SageMath
E = EllipticCurve("gw1")
E.isogeny_class()
Elliptic curves in class 80850.gw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.gw1 | 80850gm4 | \([1, 0, 0, -31903313, -69356252883]\) | \(1953542217204454969/170843779260\) | \(314056246658745937500\) | \([2]\) | \(5898240\) | \(2.9743\) | |
| 80850.gw2 | 80850gm3 | \([1, 0, 0, -11568313, 14369312117]\) | \(93137706732176569/5369647977540\) | \(9870839295462554062500\) | \([2]\) | \(5898240\) | \(2.9743\) | |
| 80850.gw3 | 80850gm2 | \([1, 0, 0, -2135813, -920770383]\) | \(586145095611769/140040608400\) | \(257431836525806250000\) | \([2, 2]\) | \(2949120\) | \(2.6277\) | |
| 80850.gw4 | 80850gm1 | \([1, 0, 0, 314187, -90220383]\) | \(1865864036231/2993760000\) | \(-5503326097500000000\) | \([2]\) | \(1474560\) | \(2.2812\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80850.gw have rank \(1\).
Complex multiplication
The elliptic curves in class 80850.gw do not have complex multiplication.Modular form 80850.2.a.gw
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
