Properties

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

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 142296i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
142296.cu4 142296i1 \([0, 1, 0, -43479, 22626162]\) \(-2725888/64827\) \(-216182362144473648\) \([2]\) \(1658880\) \(2.0069\) \(\Gamma_0(N)\)-optimal
142296.cu3 142296i2 \([0, 1, 0, -1496084, 700702176]\) \(6940769488/35721\) \(1905934294824747264\) \([2, 2]\) \(3317760\) \(2.3535\)  
142296.cu1 142296i3 \([0, 1, 0, -23907704, 44986063296]\) \(7080974546692/189\) \(40337233752904704\) \([2]\) \(6635520\) \(2.7001\)  
142296.cu2 142296i4 \([0, 1, 0, -2326144, -164552368]\) \(6522128932/3720087\) \(793957771958423288832\) \([2]\) \(6635520\) \(2.7001\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 142296.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} + 6 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.