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Properties

Label 324870n
Number of curves $4$
Conductor $324870$
CM no
Rank $1$
Graph

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

Elliptic curves in class 324870n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.n4 324870n1 \([1, 1, 0, -7228, 10742992]\) \(-355045312441/423817605120\) \(-49861717424762880\) \([2]\) \(3833856\) \(1.8830\) \(\Gamma_0(N)\)-optimal
324870.n3 324870n2 \([1, 1, 0, -775548, 259525008]\) \(438492726255762361/5824113422400\) \(685201120031937600\) \([2, 2]\) \(7667712\) \(2.2296\)  
324870.n1 324870n3 \([1, 1, 0, -12368948, 16738383768]\) \(1778827431186206888761/320129311320\) \(37662893347486680\) \([2]\) \(15335424\) \(2.5762\)  
324870.n2 324870n4 \([1, 1, 0, -1475268, -281918328]\) \(3018204753446708281/1456079336985000\) \(171306277916948265000\) \([2]\) \(15335424\) \(2.5762\)  

Rank

sage: E.rank()
 

The elliptic curves in class 324870n have rank \(1\).

Complex multiplication

The elliptic curves in class 324870n do not have complex multiplication.

Modular form 324870.2.a.n

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - q^{13} + q^{15} + q^{16} + q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.