Properties

Label 42432bn
Number of curves $4$
Conductor $42432$
CM no
Rank $1$
Graph

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

Elliptic curves in class 42432bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42432.bd4 42432bn1 \([0, -1, 0, -157, -1955]\) \(-420616192/1456611\) \(-1491569664\) \([2]\) \(24576\) \(0.44630\) \(\Gamma_0(N)\)-optimal
42432.bd3 42432bn2 \([0, -1, 0, -3537, -79695]\) \(298766385232/439569\) \(7201898496\) \([2, 2]\) \(49152\) \(0.79287\)  
42432.bd2 42432bn3 \([0, -1, 0, -4577, -27903]\) \(161838334948/87947613\) \(5763734765568\) \([2]\) \(98304\) \(1.1394\)  
42432.bd1 42432bn4 \([0, -1, 0, -56577, -5160927]\) \(305612563186948/663\) \(43450368\) \([2]\) \(98304\) \(1.1394\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42432bn have rank \(1\).

Complex multiplication

The elliptic curves in class 42432bn do not have complex multiplication.

Modular form 42432.2.a.bn

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 4 q^{11} - q^{13} - 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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 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.