Properties

Label 770.f
Number of curves $4$
Conductor $770$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 770.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.f1 770f4 \([1, 0, 0, -26116, -1580604]\) \(1969902499564819009/63690429687500\) \(63690429687500\) \([2]\) \(3456\) \(1.4227\)  
770.f2 770f2 \([1, 0, 0, -3576, 81280]\) \(5057359576472449/51765560000\) \(51765560000\) \([6]\) \(1152\) \(0.87336\)  
770.f3 770f1 \([1, 0, 0, -56, 3136]\) \(-19443408769/4249907200\) \(-4249907200\) \([6]\) \(576\) \(0.52678\) \(\Gamma_0(N)\)-optimal
770.f4 770f3 \([1, 0, 0, 504, -84560]\) \(14156681599871/3100231750000\) \(-3100231750000\) \([2]\) \(1728\) \(1.0761\)  

Rank

sage: E.rank()
 

The elliptic curves in class 770.f have rank \(1\).

Complex multiplication

The elliptic curves in class 770.f do not have complex multiplication.

Modular form 770.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{11} - 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.