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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 72828.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72828.k1 | 72828l4 | \([0, 0, 0, -4755495, 3991547198]\) | \(2640279346000/3087\) | \(13905853553071872\) | \([2]\) | \(1492992\) | \(2.3807\) | |
72828.k2 | 72828l3 | \([0, 0, 0, -294780, 63441569]\) | \(-10061824000/352947\) | \(-99368911847992752\) | \([2]\) | \(746496\) | \(2.0341\) | |
72828.k3 | 72828l2 | \([0, 0, 0, -73695, 2466326]\) | \(9826000/5103\) | \(22987227302016768\) | \([2]\) | \(497664\) | \(1.8314\) | |
72828.k4 | 72828l1 | \([0, 0, 0, 17340, 299693]\) | \(2048000/1323\) | \(-372478220171568\) | \([2]\) | \(248832\) | \(1.4848\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72828.k have rank \(1\).
Complex multiplication
The elliptic curves in class 72828.k do not have complex multiplication.Modular form 72828.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.