Properties

Label 1710h
Number of curves $4$
Conductor $1710$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1710h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.i3 1710h1 \([1, -1, 0, -279, -1715]\) \(3301293169/22800\) \(16621200\) \([2]\) \(512\) \(0.21908\) \(\Gamma_0(N)\)-optimal
1710.i2 1710h2 \([1, -1, 0, -459, 913]\) \(14688124849/8122500\) \(5921302500\) \([2, 2]\) \(1024\) \(0.56565\)  
1710.i1 1710h3 \([1, -1, 0, -5589, 161995]\) \(26487576322129/44531250\) \(32463281250\) \([2]\) \(2048\) \(0.91222\)  
1710.i4 1710h4 \([1, -1, 0, 1791, 5863]\) \(871257511151/527800050\) \(-384766236450\) \([2]\) \(2048\) \(0.91222\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1710h have rank \(1\).

Complex multiplication

The elliptic curves in class 1710h do not have complex multiplication.

Modular form 1710.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{16} - 2q^{17} - q^{19} + O(q^{20})\)  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.