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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 84042l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84042.f4 | 84042l1 | \([1, -1, 0, -405168, -100078336]\) | \(-10090256344188054273/107965577101312\) | \(-78706905706856448\) | \([2]\) | \(1228800\) | \(2.0584\) | \(\Gamma_0(N)\)-optimal |
| 84042.f3 | 84042l2 | \([1, -1, 0, -6499248, -6375761920]\) | \(41647175116728660507393/4693358285056\) | \(3421458189805824\) | \([2, 2]\) | \(2457600\) | \(2.4050\) | |
| 84042.f2 | 84042l3 | \([1, -1, 0, -6515808, -6341625136]\) | \(41966336340198080824833/442001722607124848\) | \(322219255780594014192\) | \([2]\) | \(4915200\) | \(2.7515\) | |
| 84042.f1 | 84042l4 | \([1, -1, 0, -103987968, -408126777040]\) | \(170586815436843383543017473/2166416\) | \(1579317264\) | \([2]\) | \(4915200\) | \(2.7515\) |
Rank
sage: E.rank()
The elliptic curves in class 84042l have rank \(0\).
Complex multiplication
The elliptic curves in class 84042l do not have complex multiplication.Modular form 84042.2.a.l
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
