Properties

Label 481338.df
Number of curves $4$
Conductor $481338$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("df1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 481338.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481338.df1 481338df4 \([1, -1, 1, -2733012273839, -1739042218942772169]\) \(1748094148784980747354970849498497/887694600425282263291392\) \(1146429142703505820824657281422848\) \([2]\) \(9077391360\) \(5.6763\)  
481338.df2 481338df3 \([1, -1, 1, -373858430639, 48387423133967415]\) \(4474676144192042711273397261697/1806328356954994499451382272\) \(2332815214503773770612648277062445568\) \([2]\) \(9077391360\) \(5.6763\) \(\Gamma_0(N)\)-optimal*
481338.df3 481338df2 \([1, -1, 1, -171737242799, -26863669522941897]\) \(433744050935826360922067531137/9612122270219882316693504\) \(12413748026100540598959174406373376\) \([2, 2]\) \(4538695680\) \(5.3297\) \(\Gamma_0(N)\)-optimal*
481338.df4 481338df1 \([1, -1, 1, 975020881, -1286710394570697]\) \(79374649975090937760383/553856914190911653543936\) \(-715288464086743951460818678185984\) \([2]\) \(2269347840\) \(4.9831\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 481338.df1.

Rank

sage: E.rank()
 

The elliptic curves in class 481338.df have rank \(0\).

Complex multiplication

The elliptic curves in class 481338.df do not have complex multiplication.

Modular form 481338.2.a.df

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2q^{5} + q^{8} + 2q^{10} - q^{13} + q^{16} - q^{17} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.