Properties

Label 229320.ds
Number of curves $4$
Conductor $229320$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ds1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 229320.ds

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
229320.ds1 229320o4 \([0, 0, 0, -247665747, -1500192649186]\) \(19129597231400697604/26325\) \(2311980170572800\) \([2]\) \(14155776\) \(3.1144\)  
229320.ds2 229320o2 \([0, 0, 0, -15479247, -23440071886]\) \(18681746265374416/693005625\) \(15215719497582240000\) \([2, 2]\) \(7077888\) \(2.7678\)  
229320.ds3 229320o3 \([0, 0, 0, -14764827, -25701496954]\) \(-4053153720264484/903687890625\) \(-79365944292944400000000\) \([2]\) \(14155776\) \(3.1144\)  
229320.ds4 229320o1 \([0, 0, 0, -1012242, -330478099]\) \(83587439220736/13990184325\) \(19198141466084053200\) \([2]\) \(3538944\) \(2.4212\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 229320.ds have rank \(1\).

Complex multiplication

The elliptic curves in class 229320.ds do not have complex multiplication.

Modular form 229320.2.a.ds

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.