Properties

Label 435344.bc
Number of curves $4$
Conductor $435344$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 435344.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.bc1 435344bc3 \([0, 0, 0, -2678819, 499567042]\) \(215062038362754/113550802729\) \(1122484298894462486528\) \([2]\) \(12042240\) \(2.7307\) \(\Gamma_0(N)\)-optimal*
435344.bc2 435344bc2 \([0, 0, 0, -1536379, -727185030]\) \(81144432781668/740329681\) \(3659192286431159296\) \([2, 2]\) \(6021120\) \(2.3841\) \(\Gamma_0(N)\)-optimal*
435344.bc3 435344bc1 \([0, 0, 0, -1532999, -730568410]\) \(322440248841552/27209\) \(33621157396736\) \([2]\) \(3010560\) \(2.0376\) \(\Gamma_0(N)\)-optimal*
435344.bc4 435344bc4 \([0, 0, 0, -448019, -1737400782]\) \(-1006057824354/131332646081\) \(-1298263240903855163392\) \([2]\) \(12042240\) \(2.7307\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 435344.bc1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344.bc have rank \(0\).

Complex multiplication

The elliptic curves in class 435344.bc do not have complex multiplication.

Modular form 435344.2.a.bc

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