Properties

Label 265200fk
Number of curves $4$
Conductor $265200$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 265200fk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
265200.fk3 265200fk1 \([0, 1, 0, -11908, -431812]\) \(46689225424/7249905\) \(28999620000000\) \([2]\) \(589824\) \(1.3064\) \(\Gamma_0(N)\)-optimal
265200.fk2 265200fk2 \([0, 1, 0, -52408, 4185188]\) \(994958062276/98903025\) \(1582448400000000\) \([2, 2]\) \(1179648\) \(1.6530\)  
265200.fk1 265200fk3 \([0, 1, 0, -817408, 284175188]\) \(1887517194957938/21849165\) \(699173280000000\) \([2]\) \(2359296\) \(1.9996\)  
265200.fk4 265200fk4 \([0, 1, 0, 64592, 20331188]\) \(931329171502/6107473125\) \(-195439140000000000\) \([2]\) \(2359296\) \(1.9996\)  

Rank

sage: E.rank()
 

The elliptic curves in class 265200fk have rank \(1\).

Complex multiplication

The elliptic curves in class 265200fk do not have complex multiplication.

Modular form 265200.2.a.fk

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{9} - 4q^{11} + q^{13} - q^{17} - 8q^{19} + O(q^{20})\)  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.