Properties

Label 333270.n
Number of curves $8$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.n1 333270n7 \([1, -1, 0, -30713310, 41225237550]\) \(29689921233686449/10380965400750\) \(1120294717056357019710750\) \([2]\) \(58392576\) \(3.3161\)  
333270.n2 333270n4 \([1, -1, 0, -27428220, 55296592056]\) \(21145699168383889/2593080\) \(279840430322079480\) \([2]\) \(19464192\) \(2.7668\)  
333270.n3 333270n6 \([1, -1, 0, -12859560, -17274359700]\) \(2179252305146449/66177562500\) \(7141760982178070062500\) \([2, 2]\) \(29196288\) \(2.9695\)  
333270.n4 333270n3 \([1, -1, 0, -12764340, -17549564544]\) \(2131200347946769/2058000\) \(222095579620698000\) \([2]\) \(14598144\) \(2.6229\)  
333270.n5 333270n2 \([1, -1, 0, -1718820, 859508496]\) \(5203798902289/57153600\) \(6167911525466241600\) \([2, 2]\) \(9732096\) \(2.4202\)  
333270.n6 333270n5 \([1, -1, 0, -385740, 2157661800]\) \(-58818484369/18600435000\) \(-2007324777707540235000\) \([2]\) \(19464192\) \(2.7668\)  
333270.n7 333270n1 \([1, -1, 0, -195300, -11640240]\) \(7633736209/3870720\) \(417720992200888320\) \([2]\) \(4866048\) \(2.0736\) \(\Gamma_0(N)\)-optimal
333270.n8 333270n8 \([1, -1, 0, 3470670, -58161989574]\) \(42841933504271/13565917968750\) \(-1464008947695030761718750\) \([2]\) \(58392576\) \(3.3161\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.n

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

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.