Properties

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

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 1710.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.e1 1710g3 \([1, -1, 0, -4432320, -3590548484]\) \(13209596798923694545921/92340\) \(67315860\) \([2]\) \(30720\) \(2.0368\)  
1710.e2 1710g4 \([1, -1, 0, -280440, -54592700]\) \(3345930611358906241/165622259047500\) \(120738626845627500\) \([2]\) \(30720\) \(2.0368\)  
1710.e3 1710g2 \([1, -1, 0, -277020, -56050304]\) \(3225005357698077121/8526675600\) \(6215946512400\) \([2, 2]\) \(15360\) \(1.6902\)  
1710.e4 1710g1 \([1, -1, 0, -17100, -895280]\) \(-758575480593601/40535043840\) \(-29550046959360\) \([2]\) \(7680\) \(1.3436\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1710.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + q^{10} - 6 q^{13} - 4 q^{14} + q^{16} - 2 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.