Properties

Label 142296f
Number of curves $4$
Conductor $142296$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 142296f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
142296.cm3 142296f1 \([0, 1, 0, -3480158964, -79022926981440]\) \(87364831012240243408/1760913\) \(93955501718953281792\) \([2]\) \(66355200\) \(3.8159\) \(\Gamma_0(N)\)-optimal
142296.cm2 142296f2 \([0, 1, 0, -3480277544, -79017272707584]\) \(21843440425782779332/3100814593569\) \(661789857593708721200784384\) \([2, 2]\) \(132710400\) \(4.1625\)  
142296.cm1 142296f3 \([0, 1, 0, -3795937504, -63829483128160]\) \(14171198121996897746/4077720290568771\) \(1740570968673386314456270067712\) \([2]\) \(265420800\) \(4.5090\)  
142296.cm4 142296f4 \([0, 1, 0, -3166514864, -93843186862944]\) \(-8226100326647904626/4152140742401883\) \(-1772337266679619373906549938176\) \([2]\) \(265420800\) \(4.5090\)  

Rank

sage: E.rank()
 

The elliptic curves in class 142296f have rank \(1\).

Complex multiplication

The elliptic curves in class 142296f do not have complex multiplication.

Modular form 142296.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} + q^{9} - 6q^{13} - 2q^{15} - 2q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.