Properties

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

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Show commands: SageMath
E = EllipticCurve("ic1")
 
E.isogeny_class()
 

Elliptic curves in class 235200.ic

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.ic1 235200ic4 \([0, -1, 0, -18308033, -30145452063]\) \(5633270409316/14175\) \(1707698764800000000\) \([2]\) \(9437184\) \(2.7367\)  
235200.ic2 235200ic3 \([0, -1, 0, -3216033, 1625951937]\) \(30534944836/8203125\) \(988251600000000000000\) \([2]\) \(9437184\) \(2.7367\)  
235200.ic3 235200ic2 \([0, -1, 0, -1158033, -458802063]\) \(5702413264/275625\) \(8301313440000000000\) \([2, 2]\) \(4718592\) \(2.3901\)  
235200.ic4 235200ic1 \([0, -1, 0, 42467, -27822563]\) \(4499456/180075\) \(-338970298800000000\) \([2]\) \(2359296\) \(2.0436\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 235200.2.a.ic

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