Properties

Label 162288.cf
Number of curves $4$
Conductor $162288$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 162288.cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162288.cf1 162288v3 \([0, 0, 0, -1261958691, 17255034101410]\) \(632678989847546725777/80515134\) \(28284808089498476544\) \([2]\) \(35389440\) \(3.5924\)  
162288.cf2 162288v4 \([0, 0, 0, -90239331, 186843563746]\) \(231331938231569617/90942310746882\) \(31947854755990249409421312\) \([2]\) \(35389440\) \(3.5924\)  
162288.cf3 162288v2 \([0, 0, 0, -78879171, 269561432770]\) \(154502321244119857/55101928644\) \(19357199070882488008704\) \([2, 2]\) \(17694720\) \(3.2458\)  
162288.cf4 162288v1 \([0, 0, 0, -4226691, 5455889026]\) \(-23771111713777/22848457968\) \(-8026618309619311116288\) \([2]\) \(8847360\) \(2.8993\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 162288.cf have rank \(1\).

Complex multiplication

The elliptic curves in class 162288.cf do not have complex multiplication.

Modular form 162288.2.a.cf

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