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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5040.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.j1 | 5040i5 | \([0, 0, 0, -219603, -39600398]\) | \(784478485879202/221484375\) | \(330674400000000\) | \([2]\) | \(32768\) | \(1.7671\) | |
| 5040.j2 | 5040i3 | \([0, 0, 0, -15483, -450182]\) | \(549871953124/200930625\) | \(149993907840000\) | \([2, 2]\) | \(16384\) | \(1.4205\) | |
| 5040.j3 | 5040i2 | \([0, 0, 0, -6663, 204262]\) | \(175293437776/4862025\) | \(907370553600\) | \([2, 2]\) | \(8192\) | \(1.0739\) | |
| 5040.j4 | 5040i1 | \([0, 0, 0, -6618, 207223]\) | \(2748251600896/2205\) | \(25719120\) | \([2]\) | \(4096\) | \(0.72733\) | \(\Gamma_0(N)\)-optimal |
| 5040.j5 | 5040i4 | \([0, 0, 0, 1437, 669202]\) | \(439608956/259416045\) | \(-193653039928320\) | \([2]\) | \(16384\) | \(1.4205\) | |
| 5040.j6 | 5040i6 | \([0, 0, 0, 47517, -3184382]\) | \(7947184069438/7533176175\) | \(-11246971763865600\) | \([2]\) | \(32768\) | \(1.7671\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.j have rank \(0\).
Complex multiplication
The elliptic curves in class 5040.j do not have complex multiplication.Modular form 5040.2.a.j
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.
