Properties

Label 297689.b
Number of curves $4$
Conductor $297689$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 297689.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
297689.b1 297689b4 \([1, -1, 0, -228698, -42029359]\) \(209267191953/55223\) \(349084631654927\) \([2]\) \(1505280\) \(1.7750\)  
297689.b2 297689b2 \([1, -1, 0, -16063, -480480]\) \(72511713/25921\) \(163856051593129\) \([2, 2]\) \(752640\) \(1.4284\)  
297689.b3 297689b1 \([1, -1, 0, -6818, 212895]\) \(5545233/161\) \(1017739450889\) \([2]\) \(376320\) \(1.0818\) \(\Gamma_0(N)\)-optimal
297689.b4 297689b3 \([1, -1, 0, 48652, -3418541]\) \(2014698447/1958887\) \(-12382835898966463\) \([2]\) \(1505280\) \(1.7750\)  

Rank

sage: E.rank()
 

The elliptic curves in class 297689.b have rank \(0\).

Complex multiplication

The elliptic curves in class 297689.b do not have complex multiplication.

Modular form 297689.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - 2 q^{5} - q^{7} - 3 q^{8} - 3 q^{9} - 2 q^{10} + 4 q^{11} + 6 q^{13} - q^{14} - q^{16} - 2 q^{17} - 3 q^{18} - 4 q^{19} + O(q^{20})\) 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.