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Properties

Label 463680.dd
Number of curves $4$
Conductor $463680$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("dd1")
 
E.isogeny_class()
 

Elliptic curves in class 463680.dd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.dd1 463680dd4 \([0, 0, 0, -1061868, -393235792]\) \(692895692874169/51420783750\) \(9826664802877440000\) \([2]\) \(9437184\) \(2.3900\)  
463680.dd2 463680dd2 \([0, 0, 0, -215148, 31140272]\) \(5763259856089/1143116100\) \(218452888623513600\) \([2, 2]\) \(4718592\) \(2.0435\)  
463680.dd3 463680dd1 \([0, 0, 0, -203628, 35365808]\) \(4886171981209/270480\) \(51689532948480\) \([2]\) \(2359296\) \(1.6969\) \(\Gamma_0(N)\)-optimal*
463680.dd4 463680dd3 \([0, 0, 0, 447252, 185082032]\) \(51774168853511/107398242630\) \(-20524123783763066880\) \([2]\) \(9437184\) \(2.3900\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 463680.dd1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.dd have rank \(0\).

Complex multiplication

The elliptic curves in class 463680.dd do not have complex multiplication.

Modular form 463680.2.a.dd

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