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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 32912.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32912.i1 | 32912t4 | \([0, 0, 0, -175571, -28315694]\) | \(82483294977/17\) | \(123357335552\) | \([2]\) | \(92160\) | \(1.5155\) | |
32912.i2 | 32912t2 | \([0, 0, 0, -11011, -439230]\) | \(20346417/289\) | \(2097074704384\) | \([2, 2]\) | \(46080\) | \(1.1689\) | |
32912.i3 | 32912t3 | \([0, 0, 0, -1331, -1184590]\) | \(-35937/83521\) | \(-606054589566976\) | \([2]\) | \(92160\) | \(1.5155\) | |
32912.i4 | 32912t1 | \([0, 0, 0, -1331, 7986]\) | \(35937/17\) | \(123357335552\) | \([2]\) | \(23040\) | \(0.82231\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32912.i have rank \(1\).
Complex multiplication
The elliptic curves in class 32912.i do not have complex multiplication.Modular form 32912.2.a.i
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.