Properties

Label 333270.o
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.o1 333270o3 \([1, -1, 0, -467606475, 3892072260325]\) \(104778147797811105409/289854482400\) \(31280563301402044274400\) \([2]\) \(86507520\) \(3.5492\)  
333270.o2 333270o2 \([1, -1, 0, -29594475, 59204453125]\) \(26562019806177409/1343744640000\) \(145014453198739635840000\) \([2, 2]\) \(43253760\) \(3.2026\)  
333270.o3 333270o1 \([1, -1, 0, -5218155, -3398811899]\) \(145606291302529/37984665600\) \(4099235336798050713600\) \([2]\) \(21626880\) \(2.8560\) \(\Gamma_0(N)\)-optimal
333270.o4 333270o4 \([1, -1, 0, 18396405, 232653091621]\) \(6380108151242111/220374787500000\) \(-23782442256365720287500000\) \([2]\) \(86507520\) \(3.5492\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.o have rank \(1\).

Complex multiplication

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

Modular form 333270.2.a.o

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.