Properties

Label 68400.fi
Number of curves $4$
Conductor $68400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fi1")
 
E.isogeny_class()
 

Elliptic curves in class 68400.fi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68400.fi1 68400ej4 \([0, 0, 0, -1667631675, 16737529734250]\) \(10993009831928446009969/3767761230468750000\) \(175788667968750000000000000000\) \([2]\) \(79626240\) \(4.3135\)  
68400.fi2 68400ej2 \([0, 0, 0, -1493967675, 22225926438250]\) \(7903870428425797297009/886464000000\) \(41358864384000000000000\) \([2]\) \(26542080\) \(3.7642\)  
68400.fi3 68400ej1 \([0, 0, 0, -93135675, 349133094250]\) \(-1914980734749238129/20440940544000\) \(-953692522020864000000000\) \([2]\) \(13271040\) \(3.4177\) \(\Gamma_0(N)\)-optimal
68400.fi4 68400ej3 \([0, 0, 0, 307760325, 1817393958250]\) \(69096190760262356111/70568821500000000\) \(-3292458935904000000000000000\) \([2]\) \(39813120\) \(3.9670\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68400.fi have rank \(1\).

Complex multiplication

The elliptic curves in class 68400.fi do not have complex multiplication.

Modular form 68400.2.a.fi

sage: E.q_eigenform(10)
 
\(q + 2 q^{7} + 6 q^{11} + 4 q^{13} - 6 q^{17} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.