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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 274890.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 274890.ew1 | 274890ew5 | \([1, 0, 0, -14314861, 20842617935]\) | \(2757381641970898311361/379829992662450\) | \(44686618806744580050\) | \([2]\) | \(18874368\) | \(2.7881\) | |
| 274890.ew2 | 274890ew3 | \([1, 0, 0, -974611, 263948285]\) | \(870220733067747361/247623269602500\) | \(29132630045464522500\) | \([2, 2]\) | \(9437184\) | \(2.4415\) | |
| 274890.ew3 | 274890ew2 | \([1, 0, 0, -362111, -80644215]\) | \(44633474953947361/1967006250000\) | \(231416318306250000\) | \([2, 2]\) | \(4718592\) | \(2.0950\) | |
| 274890.ew4 | 274890ew1 | \([1, 0, 0, -358191, -82542279]\) | \(43199583152847841/89760000\) | \(10560174240000\) | \([2]\) | \(2359296\) | \(1.7484\) | \(\Gamma_0(N)\)-optimal |
| 274890.ew5 | 274890ew4 | \([1, 0, 0, 187669, -303744939]\) | \(6213165856218719/342407226562500\) | \(-40283867797851562500\) | \([2]\) | \(9437184\) | \(2.4415\) | |
| 274890.ew6 | 274890ew6 | \([1, 0, 0, 2565639, 1741648635]\) | \(15875306080318016639/20322604533582450\) | \(-2390934100771441660050\) | \([2]\) | \(18874368\) | \(2.7881\) |
Rank
sage: E.rank()
The elliptic curves in class 274890.ew have rank \(0\).
Complex multiplication
The elliptic curves in class 274890.ew do not have complex multiplication.Modular form 274890.2.a.ew
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
