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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4200k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.z4 | 4200k1 | \([0, 1, 0, -18383, 953238]\) | \(2748251600896/2205\) | \(551250000\) | \([4]\) | \(6144\) | \(0.98275\) | \(\Gamma_0(N)\)-optimal |
| 4200.z3 | 4200k2 | \([0, 1, 0, -18508, 939488]\) | \(175293437776/4862025\) | \(19448100000000\) | \([2, 2]\) | \(12288\) | \(1.3293\) | |
| 4200.z2 | 4200k3 | \([0, 1, 0, -43008, -2098512]\) | \(549871953124/200930625\) | \(3214890000000000\) | \([2, 2]\) | \(24576\) | \(1.6759\) | |
| 4200.z5 | 4200k4 | \([0, 1, 0, 3992, 3099488]\) | \(439608956/259416045\) | \(-4150656720000000\) | \([2]\) | \(24576\) | \(1.6759\) | |
| 4200.z1 | 4200k5 | \([0, 1, 0, -610008, -183538512]\) | \(784478485879202/221484375\) | \(7087500000000000\) | \([2]\) | \(49152\) | \(2.0225\) | |
| 4200.z6 | 4200k6 | \([0, 1, 0, 131992, -14698512]\) | \(7947184069438/7533176175\) | \(-241061637600000000\) | \([2]\) | \(49152\) | \(2.0225\) |
Rank
sage: E.rank()
The elliptic curves in class 4200k have rank \(0\).
Complex multiplication
The elliptic curves in class 4200k do not have complex multiplication.Modular form 4200.2.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 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.
