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Properties

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

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Show commands: SageMath
E = EllipticCurve("bn1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.bn1 324870bn4 \([1, 0, 1, -46614069, -122500368728]\) \(95210863233510962017081/1206641250360\) \(141960136463603640\) \([2]\) \(21233664\) \(2.8520\)  
324870.bn2 324870bn2 \([1, 0, 1, -2915869, -1910816008]\) \(23304472877725373881/82743765249600\) \(9734721237850190400\) \([2, 2]\) \(10616832\) \(2.5054\)  
324870.bn3 324870bn3 \([1, 0, 1, -1616389, -3622491064]\) \(-3969837635175430201/45883867071315000\) \(-5398191077073138435000\) \([2]\) \(21233664\) \(2.8520\)  
324870.bn4 324870bn1 \([1, 0, 1, -265949, 306296]\) \(17681870665400761/10232167895040\) \(1203804320683560960\) \([2]\) \(5308416\) \(2.1588\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 324870.2.a.bn

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 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.