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Properties

Label 301665.n
Number of curves $4$
Conductor $301665$
CM no
Rank $1$
Graph

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

Elliptic curves in class 301665.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
301665.n1 301665n4 \([1, 1, 1, -70482045, -1791022680]\) \(8022303494868395314009/4642258987446136875\) \(22407297460935900483481875\) \([2]\) \(76382208\) \(3.5538\)  
301665.n2 301665n2 \([1, 1, 1, -48259390, 128620405922]\) \(2575188849443651233129/9189111800621025\) \(44354087541243769059225\) \([2, 2]\) \(38191104\) \(3.2072\)  
301665.n3 301665n1 \([1, 1, 1, -48217985, 128852837030]\) \(2568566247768320202649/32879930265\) \(158705143322474385\) \([4]\) \(19095552\) \(2.8606\) \(\Gamma_0(N)\)-optimal
301665.n4 301665n3 \([1, 1, 1, -26699215, 244157071712]\) \(-436072878965111198329/5083471030070311515\) \(-24536943719182650253405635\) \([2]\) \(76382208\) \(3.5538\)  

Rank

sage: E.rank()
 

The elliptic curves in class 301665.n have rank \(1\).

Complex multiplication

The elliptic curves in class 301665.n do not have complex multiplication.

Modular form 301665.2.a.n

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} - q^{14} - q^{15} - q^{16} - q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) 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.