Properties

Label 1617g
Number of curves $6$
Conductor $1617$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 1617g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1617.e4 1617g1 \([1, 0, 0, -1667, -26328]\) \(4354703137/1617\) \(190238433\) \([2]\) \(960\) \(0.55602\) \(\Gamma_0(N)\)-optimal
1617.e3 1617g2 \([1, 0, 0, -1912, -18145]\) \(6570725617/2614689\) \(307615546161\) \([2, 2]\) \(1920\) \(0.90260\)  
1617.e2 1617g3 \([1, 0, 0, -13917, 618120]\) \(2533811507137/58110129\) \(6836598566721\) \([2, 2]\) \(3840\) \(1.2492\)  
1617.e6 1617g4 \([1, 0, 0, 6173, -129718]\) \(221115865823/190238433\) \(-22381361404017\) \([2]\) \(3840\) \(1.2492\)  
1617.e1 1617g5 \([1, 0, 0, -221432, 40087473]\) \(10206027697760497/5557167\) \(653795140383\) \([4]\) \(7680\) \(1.5957\)  
1617.e5 1617g6 \([1, 0, 0, 1518, 1917747]\) \(3288008303/13504609503\) \(-1588803803418447\) \([2]\) \(7680\) \(1.5957\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1617g have rank \(1\).

Complex multiplication

The elliptic curves in class 1617g do not have complex multiplication.

Modular form 1617.2.a.g

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} - q^{11} - q^{12} - 6 q^{13} + 2 q^{15} - q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.