Properties

Label 47040.fh
Number of curves $4$
Conductor $47040$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fh1")
 
E.isogeny_class()
 

Elliptic curves in class 47040.fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.fh1 47040cm4 \([0, 1, 0, -63507201, 194775968799]\) \(7347751505995469192/72930375\) \(281155524636672000\) \([2]\) \(2949120\) \(2.9244\)  
47040.fh2 47040cm3 \([0, 1, 0, -5687201, 157376799]\) \(5276930158229192/3050936350875\) \(11761733164862435328000\) \([2]\) \(2949120\) \(2.9244\)  
47040.fh3 47040cm2 \([0, 1, 0, -3972201, 3037547799]\) \(14383655824793536/45209390625\) \(21785966991936000000\) \([2, 2]\) \(1474560\) \(2.5778\)  
47040.fh4 47040cm1 \([0, 1, 0, -144076, 87594674]\) \(-43927191786304/415283203125\) \(-3126889828125000000\) \([2]\) \(737280\) \(2.2313\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 47040.fh have rank \(1\).

Complex multiplication

The elliptic curves in class 47040.fh do not have complex multiplication.

Modular form 47040.2.a.fh

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.