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Properties

Label 324870.r
Number of curves $2$
Conductor $324870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.r1 324870r2 \([1, 1, 0, -68783873, 217554262773]\) \(305911187318336733511561/3193776066377594880\) \(375744560433257660037120\) \([2]\) \(74612736\) \(3.3403\)  
324870.r2 324870r1 \([1, 1, 0, -1046273, 8421196533]\) \(-1076641646065581961/259805183842713600\) \(-30565820073911412326400\) \([2]\) \(37306368\) \(2.9938\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 324870.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.