Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 6930.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6930.q1 | 6930o8 | \([1, -1, 0, -47916009, 127676162763]\) | \(16689299266861680229173649/2396798250\) | \(1747265924250\) | \([6]\) | \(331776\) | \(2.6717\) | |
6930.q2 | 6930o7 | \([1, -1, 0, -3073509, 1885066263]\) | \(4404531606962679693649/444872222400201750\) | \(324311850129747075750\) | \([6]\) | \(331776\) | \(2.6717\) | |
6930.q3 | 6930o6 | \([1, -1, 0, -2994759, 1995489513]\) | \(4074571110566294433649/48828650062500\) | \(35596085895562500\) | \([2, 6]\) | \(165888\) | \(2.3252\) | |
6930.q4 | 6930o4 | \([1, -1, 0, -675099, -212909715]\) | \(46676570542430835889/106752955783320\) | \(77822904766040280\) | \([2]\) | \(110592\) | \(2.1224\) | |
6930.q5 | 6930o5 | \([1, -1, 0, -592299, 174808125]\) | \(31522423139920199089/164434491947880\) | \(119872744630004520\) | \([2]\) | \(110592\) | \(2.1224\) | |
6930.q6 | 6930o3 | \([1, -1, 0, -182259, 32927013]\) | \(-918468938249433649/109183593750000\) | \(-79594839843750000\) | \([6]\) | \(82944\) | \(1.9786\) | |
6930.q7 | 6930o2 | \([1, -1, 0, -57699, -647595]\) | \(29141055407581489/16604321025600\) | \(12104550027662400\) | \([2, 2]\) | \(55296\) | \(1.7758\) | |
6930.q8 | 6930o1 | \([1, -1, 0, 14301, -85995]\) | \(443688652450511/260789760000\) | \(-190115735040000\) | \([2]\) | \(27648\) | \(1.4293\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6930.q have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.q do not have complex multiplication.Modular form 6930.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.