Properties

Label 85800.cy
Number of curves $4$
Conductor $85800$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 85800.cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
85800.cy1 85800cp4 \([0, 1, 0, -641521408, 6253886494688]\) \(912446049969377120252018/17177299425\) \(549673581600000000\) \([2]\) \(16515072\) \(3.3921\)  
85800.cy2 85800cp3 \([0, 1, 0, -43671408, 79238894688]\) \(287849398425814280018/81784533026485575\) \(2617105056847538400000000\) \([2]\) \(16515072\) \(3.3921\)  
85800.cy3 85800cp2 \([0, 1, 0, -40096408, 97700194688]\) \(445574312599094932036/61129333175625\) \(978069330810000000000\) \([2, 2]\) \(8257536\) \(3.0455\)  
85800.cy4 85800cp1 \([0, 1, 0, -2283908, 1807694688]\) \(-329381898333928144/162600887109375\) \(-650403548437500000000\) \([2]\) \(4128768\) \(2.6989\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 85800.cy have rank \(1\).

Complex multiplication

The elliptic curves in class 85800.cy do not have complex multiplication.

Modular form 85800.2.a.cy

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.