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SageMath
E = EllipticCurve("fq1")
E.isogeny_class()
Elliptic curves in class 58800.fq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58800.fq1 | 58800do6 | \([0, 1, 0, -29890408, -62893928812]\) | \(784478485879202/221484375\) | \(833837287500000000000\) | \([2]\) | \(4718592\) | \(2.9954\) | |
58800.fq2 | 58800do4 | \([0, 1, 0, -2107408, -715574812]\) | \(549871953124/200930625\) | \(378228593610000000000\) | \([2, 2]\) | \(2359296\) | \(2.6488\) | |
58800.fq3 | 58800do2 | \([0, 1, 0, -906908, 324058188]\) | \(175293437776/4862025\) | \(2288049516900000000\) | \([2, 2]\) | \(1179648\) | \(2.3023\) | |
58800.fq4 | 58800do1 | \([0, 1, 0, -900783, 328762188]\) | \(2748251600896/2205\) | \(64854011250000\) | \([2]\) | \(589824\) | \(1.9557\) | \(\Gamma_0(N)\)-optimal |
58800.fq5 | 58800do3 | \([0, 1, 0, 195592, 1062733188]\) | \(439608956/259416045\) | \(-488320612451280000000\) | \([2]\) | \(2359296\) | \(2.6488\) | |
58800.fq6 | 58800do5 | \([0, 1, 0, 6467592, -5054524812]\) | \(7947184069438/7533176175\) | \(-28360660602002400000000\) | \([2]\) | \(4718592\) | \(2.9954\) |
Rank
sage: E.rank()
The elliptic curves in class 58800.fq have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.fq do not have complex multiplication.Modular form 58800.2.a.fq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.