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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 18810.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.c1 | 18810d4 | \([1, -1, 0, -44061345, 111468497571]\) | \(12976854634417729473922321/148112152782766327650\) | \(107973759378636652856850\) | \([2]\) | \(2560000\) | \(3.2329\) | |
18810.c2 | 18810d2 | \([1, -1, 0, -4094595, -3187970379]\) | \(10414276373665867414321/301547812500000\) | \(219828355312500000\) | \([2]\) | \(512000\) | \(2.4281\) | |
18810.c3 | 18810d3 | \([1, -1, 0, -577575, 4420152585]\) | \(-29229525625065721201/11560253601080069820\) | \(-8427424875187370898780\) | \([2]\) | \(1280000\) | \(2.8863\) | |
18810.c4 | 18810d1 | \([1, -1, 0, -245475, -54016875]\) | \(-2243980016705847601/434411683200000\) | \(-316686117052800000\) | \([2]\) | \(256000\) | \(2.0816\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18810.c have rank \(0\).
Complex multiplication
The elliptic curves in class 18810.c do not have complex multiplication.Modular form 18810.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.