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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 61347.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61347.w1 | 61347j4 | \([1, 1, 0, -2996204, -1995493287]\) | \(347873904937/395307\) | \(3380264851525061643\) | \([2]\) | \(1658880\) | \(2.4688\) | |
61347.w2 | 61347j2 | \([1, 1, 0, -235589, -13923840]\) | \(169112377/88209\) | \(754273975133691441\) | \([2, 2]\) | \(829440\) | \(2.1222\) | |
61347.w3 | 61347j1 | \([1, 1, 0, -133344, 18528723]\) | \(30664297/297\) | \(2539643013918153\) | \([2]\) | \(414720\) | \(1.7756\) | \(\Gamma_0(N)\)-optimal |
61347.w4 | 61347j3 | \([1, 1, 0, 889106, -107273525]\) | \(9090072503/5845851\) | \(-49987793442951005499\) | \([2]\) | \(1658880\) | \(2.4688\) |
Rank
sage: E.rank()
The elliptic curves in class 61347.w have rank \(0\).
Complex multiplication
The elliptic curves in class 61347.w do not have complex multiplication.Modular form 61347.2.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.