Properties

Label 6930.q
Number of curves $8$
Conductor $6930$
CM no
Rank $1$
Graph

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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)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + q^{11} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.