Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 6006o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6006.o4 | 6006o1 | \([1, 0, 1, -309011326, 2090760854000]\) | \(3263224124812796801735447265625/1837810787484672\) | \(1837810787484672\) | \([6]\) | \(829440\) | \(3.1568\) | \(\Gamma_0(N)\)-optimal |
| 6006.o3 | 6006o2 | \([1, 0, 1, -309013086, 2090735846512]\) | \(3263279883032933444452132257625/77441472526453540753248\) | \(77441472526453540753248\) | \([6]\) | \(1658880\) | \(3.5034\) | |
| 6006.o2 | 6006o3 | \([1, 0, 1, -309623821, 2082056315144]\) | \(3282666836869681281754155591625/26942969374939856448258048\) | \(26942969374939856448258048\) | \([2]\) | \(2488320\) | \(3.7061\) | |
| 6006.o1 | 6006o4 | \([1, 0, 1, -527694861, -1230791696120]\) | \(16250708692977087048493451847625/8749977648266474863605153792\) | \(8749977648266474863605153792\) | \([2]\) | \(4976640\) | \(4.0527\) |
Rank
sage: E.rank()
The elliptic curves in class 6006o have rank \(0\).
Complex multiplication
The elliptic curves in class 6006o do not have complex multiplication.Modular form 6006.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
