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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 42042.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42042.t1 | 42042x4 | \([1, 1, 0, -29218284, -60801900060]\) | \(23447665694255643433657/3193234044\) | \(375680792042556\) | \([2]\) | \(1769472\) | \(2.6521\) | |
| 42042.t2 | 42042x3 | \([1, 1, 0, -2093844, -654042948]\) | \(8629164767308099897/3427163787379332\) | \(403202392421391030468\) | \([2]\) | \(1769472\) | \(2.6521\) | |
| 42042.t3 | 42042x2 | \([1, 1, 0, -1826304, -950423760]\) | \(5726048926423698937/2112522716304\) | \(248536185050449296\) | \([2, 2]\) | \(884736\) | \(2.3056\) | |
| 42042.t4 | 42042x1 | \([1, 1, 0, -97584, -19335168]\) | \(-873530903492857/861466814208\) | \(-101350709224756992\) | \([2]\) | \(442368\) | \(1.9590\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42042.t have rank \(0\).
Complex multiplication
The elliptic curves in class 42042.t do not have complex multiplication.Modular form 42042.2.a.t
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.
