Properties

Label 34680bc
Number of curves $6$
Conductor $34680$
CM no
Rank $0$
Graph

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

Elliptic curves in class 34680bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34680.f5 34680bc1 \([0, -1, 0, -4431, 114840]\) \(24918016/45\) \(17379049680\) \([2]\) \(36864\) \(0.85688\) \(\Gamma_0(N)\)-optimal
34680.f4 34680bc2 \([0, -1, 0, -5876, 35076]\) \(3631696/2025\) \(12512915769600\) \([2, 2]\) \(73728\) \(1.2035\)  
34680.f6 34680bc3 \([0, -1, 0, 23024, 254716]\) \(54607676/32805\) \(-810836941870080\) \([2]\) \(147456\) \(1.5500\)  
34680.f2 34680bc4 \([0, -1, 0, -57896, -5312580]\) \(868327204/5625\) \(139032397440000\) \([2, 2]\) \(147456\) \(1.5500\)  
34680.f3 34680bc5 \([0, -1, 0, -23216, -11652084]\) \(-27995042/1171875\) \(-57930165600000000\) \([2]\) \(294912\) \(1.8966\)  
34680.f1 34680bc6 \([0, -1, 0, -924896, -342055380]\) \(1770025017602/75\) \(3707530598400\) \([2]\) \(294912\) \(1.8966\)  

Rank

sage: E.rank()
 

The elliptic curves in class 34680bc have rank \(0\).

Complex multiplication

The elliptic curves in class 34680bc do not have complex multiplication.

Modular form 34680.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} + 6 q^{13} + q^{15} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.