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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 20160cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.fk4 | 20160cn1 | \([0, 0, 0, -26472, -1657784]\) | \(2748251600896/2205\) | \(1646023680\) | \([2]\) | \(32768\) | \(1.0739\) | \(\Gamma_0(N)\)-optimal |
20160.fk3 | 20160cn2 | \([0, 0, 0, -26652, -1634096]\) | \(175293437776/4862025\) | \(58071715430400\) | \([2, 2]\) | \(65536\) | \(1.4205\) | |
20160.fk2 | 20160cn3 | \([0, 0, 0, -61932, 3601456]\) | \(549871953124/200930625\) | \(9599610101760000\) | \([2, 2]\) | \(131072\) | \(1.7671\) | |
20160.fk5 | 20160cn4 | \([0, 0, 0, 5748, -5353616]\) | \(439608956/259416045\) | \(-12393794555412480\) | \([2]\) | \(131072\) | \(1.7671\) | |
20160.fk1 | 20160cn5 | \([0, 0, 0, -878412, 316803184]\) | \(784478485879202/221484375\) | \(21163161600000000\) | \([2]\) | \(262144\) | \(2.1136\) | |
20160.fk6 | 20160cn6 | \([0, 0, 0, 190068, 25475056]\) | \(7947184069438/7533176175\) | \(-719806192887398400\) | \([2]\) | \(262144\) | \(2.1136\) |
Rank
sage: E.rank()
The elliptic curves in class 20160cn have rank \(0\).
Complex multiplication
The elliptic curves in class 20160cn do not have complex multiplication.Modular form 20160.2.a.cn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.