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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 122010.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122010.w1 | 122010bf4 | \([1, 0, 1, -69410878, -222587464144]\) | \(314353338448506783273289/141150729600\) | \(16606242186710400\) | \([2]\) | \(8257536\) | \(2.8890\) | |
| 122010.w2 | 122010bf2 | \([1, 0, 1, -4338878, -3477025744]\) | \(76783454067608361289/51426109440000\) | \(6050230349506560000\) | \([2, 2]\) | \(4128768\) | \(2.5424\) | |
| 122010.w3 | 122010bf3 | \([1, 0, 1, -3492158, -4874452432]\) | \(-40032890408196055369/64068733350000000\) | \(-7537622409894150000000\) | \([2]\) | \(8257536\) | \(2.8890\) | |
| 122010.w4 | 122010bf1 | \([1, 0, 1, -324798, -31339472]\) | \(32208729120020809/15039096422400\) | \(1769334654998937600\) | \([2]\) | \(2064384\) | \(2.1959\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 122010.w have rank \(0\).
Complex multiplication
The elliptic curves in class 122010.w do not have complex multiplication.Modular form 122010.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
