Properties

Label 330330.d
Number of curves $2$
Conductor $330330$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 330330.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330330.d1 330330d2 \([1, 1, 0, -354880578208, 81371190960406912]\) \(3713576311940579708906737903766338259/521203066947867573360000\) \(693721282107611740142160000\) \([2]\) \(1966325760\) \(5.0038\)  
330330.d2 330330d1 \([1, 1, 0, -22178041888, 1271657990714368]\) \(-906390311946839196100854971335379/339659381598543082240819200\) \(-452086636907660842462530355200\) \([2]\) \(983162880\) \(4.6573\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 330330.d have rank \(1\).

Complex multiplication

The elliptic curves in class 330330.d do not have complex multiplication.

Modular form 330330.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - q^{13} + q^{14} + q^{15} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.