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SageMath
E = EllipticCurve("ik1")
E.isogeny_class()
Elliptic curves in class 388080.ik
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.ik1 | 388080ik4 | \([0, 0, 0, -54784107, -155572334694]\) | \(1917114236485083/7117764500\) | \(67512329775618545664000\) | \([2]\) | \(47775744\) | \(3.2405\) | |
388080.ik2 | 388080ik2 | \([0, 0, 0, -3557547, 2415309274]\) | \(382704614800227/27778076480\) | \(361421614426026147840\) | \([2]\) | \(15925248\) | \(2.6912\) | |
388080.ik3 | 388080ik3 | \([0, 0, 0, -1864107, -4655078694]\) | \(-75526045083/943250000\) | \(-8946770444688384000000\) | \([2]\) | \(23887872\) | \(2.8939\) | |
388080.ik4 | 388080ik1 | \([0, 0, 0, 205653, 165668314]\) | \(73929353373/954060800\) | \(-12413321521555046400\) | \([2]\) | \(7962624\) | \(2.3446\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388080.ik have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.ik do not have complex multiplication.Modular form 388080.2.a.ik
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.