Properties

Label 92778.r
Number of curves $6$
Conductor $92778$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 92778.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
92778.r1 92778m4 \([1, 1, 1, -2968942, -1970259601]\) \(268498407453697/252\) \(2716362262908\) \([2]\) \(1695744\) \(2.1142\)  
92778.r2 92778m6 \([1, 1, 1, -2019072, 1092828159]\) \(84448510979617/933897762\) \(10066685071869193698\) \([2]\) \(3391488\) \(2.4607\)  
92778.r3 92778m3 \([1, 1, 1, -229782, -15100209]\) \(124475734657/63011844\) \(679218234753356676\) \([2, 2]\) \(1695744\) \(2.1142\)  
92778.r4 92778m2 \([1, 1, 1, -185602, -30828289]\) \(65597103937/63504\) \(684523290252816\) \([2, 2]\) \(847872\) \(1.7676\)  
92778.r5 92778m1 \([1, 1, 1, -8882, -715201]\) \(-7189057/16128\) \(-173847184826112\) \([2]\) \(423936\) \(1.4210\) \(\Gamma_0(N)\)-optimal
92778.r6 92778m5 \([1, 1, 1, 852628, -115547857]\) \(6359387729183/4218578658\) \(-45472967736905848482\) \([2]\) \(3391488\) \(2.4607\)  

Rank

sage: E.rank()
 

The elliptic curves in class 92778.r have rank \(1\).

Complex multiplication

The elliptic curves in class 92778.r do not have complex multiplication.

Modular form 92778.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - 6 q^{13} - q^{14} - 2 q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.