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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 53235bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53235.bg3 | 53235bc1 | \([1, -1, 0, -3834, -85617]\) | \(1771561/105\) | \(369468094905\) | \([2]\) | \(61440\) | \(0.97267\) | \(\Gamma_0(N)\)-optimal |
53235.bg2 | 53235bc2 | \([1, -1, 0, -11439, 366120]\) | \(47045881/11025\) | \(38794149965025\) | \([2, 2]\) | \(122880\) | \(1.3192\) | |
53235.bg4 | 53235bc3 | \([1, -1, 0, 26586, 2259765]\) | \(590589719/972405\) | \(-3421644026915205\) | \([2]\) | \(245760\) | \(1.6658\) | |
53235.bg1 | 53235bc4 | \([1, -1, 0, -171144, 27292383]\) | \(157551496201/13125\) | \(46183511863125\) | \([2]\) | \(245760\) | \(1.6658\) |
Rank
sage: E.rank()
The elliptic curves in class 53235bc have rank \(1\).
Complex multiplication
The elliptic curves in class 53235bc do not have complex multiplication.Modular form 53235.2.a.bc
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 Cremona 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 Cremona labels.