Properties

Label 1170i
Number of curves $4$
Conductor $1170$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 1170i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1170.h4 1170i1 \([1, -1, 1, 97, -313]\) \(3774555693/3515200\) \(-94910400\) \([6]\) \(576\) \(0.21793\) \(\Gamma_0(N)\)-optimal
1170.h3 1170i2 \([1, -1, 1, -503, -2473]\) \(520300455507/193072360\) \(5212953720\) \([6]\) \(1152\) \(0.56450\)  
1170.h2 1170i3 \([1, -1, 1, -2243, -40769]\) \(-63378025803/812500\) \(-15992437500\) \([2]\) \(1728\) \(0.76723\)  
1170.h1 1170i4 \([1, -1, 1, -35993, -2619269]\) \(261984288445803/42250\) \(831606750\) \([2]\) \(3456\) \(1.1138\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1170i have rank \(1\).

Complex multiplication

The elliptic curves in class 1170i do not have complex multiplication.

Modular form 1170.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.