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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 235200.ic
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.ic1 | 235200ic4 | \([0, -1, 0, -18308033, -30145452063]\) | \(5633270409316/14175\) | \(1707698764800000000\) | \([2]\) | \(9437184\) | \(2.7367\) | |
235200.ic2 | 235200ic3 | \([0, -1, 0, -3216033, 1625951937]\) | \(30534944836/8203125\) | \(988251600000000000000\) | \([2]\) | \(9437184\) | \(2.7367\) | |
235200.ic3 | 235200ic2 | \([0, -1, 0, -1158033, -458802063]\) | \(5702413264/275625\) | \(8301313440000000000\) | \([2, 2]\) | \(4718592\) | \(2.3901\) | |
235200.ic4 | 235200ic1 | \([0, -1, 0, 42467, -27822563]\) | \(4499456/180075\) | \(-338970298800000000\) | \([2]\) | \(2359296\) | \(2.0436\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235200.ic have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.ic do not have complex multiplication.Modular form 235200.2.a.ic
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.