Properties

Label 333270.db
Number of curves $4$
Conductor $333270$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("db1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.db

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.db1 333270db4 \([1, -1, 1, -14352789218, 1819214009057]\) \(3029968325354577848895529/1753440696000000000000\) \(189228098983790144376000000000000\) \([2]\) \(1167851520\) \(4.8828\)  
333270.db2 333270db2 \([1, -1, 1, -9873569003, 377624175754331]\) \(986396822567235411402169/6336721794060000\) \(683847375970293977098860000\) \([2]\) \(389283840\) \(4.3335\)  
333270.db3 333270db1 \([1, -1, 1, -605235083, 6138230240027]\) \(-227196402372228188089/19338934824115200\) \(-2087022322161694205130931200\) \([2]\) \(194641920\) \(3.9870\) \(\Gamma_0(N)\)-optimal
333270.db4 333270db3 \([1, -1, 1, 3588182302, 226055738081]\) \(47342661265381757089751/27397579603968000000\) \(-2956696463725698018705408000000\) \([2]\) \(583925760\) \(4.5363\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.db have rank \(0\).

Complex multiplication

The elliptic curves in class 333270.db do not have complex multiplication.

Modular form 333270.2.a.db

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 4q^{13} - q^{14} + q^{16} - 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.