Properties

Label 235200qp
Number of curves $4$
Conductor $235200$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("qp1")
 
E.isogeny_class()
 

Elliptic curves in class 235200qp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.qp4 235200qp1 \([0, 1, 0, 53492, 2615738]\) \(143877824/108045\) \(-12711386205000000\) \([2]\) \(1179648\) \(1.7783\) \(\Gamma_0(N)\)-optimal
235200.qp3 235200qp2 \([0, 1, 0, -246633, 22123863]\) \(220348864/99225\) \(747118209600000000\) \([2, 2]\) \(2359296\) \(2.1249\)  
235200.qp1 235200qp3 \([0, 1, 0, -3333633, 2340460863]\) \(68017239368/39375\) \(2371803840000000000\) \([2]\) \(4718592\) \(2.4714\)  
235200.qp2 235200qp4 \([0, 1, 0, -1961633, -1042891137]\) \(13858588808/229635\) \(13832359994880000000\) \([2]\) \(4718592\) \(2.4714\)  

Rank

sage: E.rank()
 

The elliptic curves in class 235200qp have rank \(1\).

Complex multiplication

The elliptic curves in class 235200qp do not have complex multiplication.

Modular form 235200.2.a.qp

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