Properties

Label 770.c
Number of curves $4$
Conductor $770$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 770.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.c1 770c3 \([1, -1, 0, -3266669, -2271693567]\) \(3855131356812007128171561/8967612500\) \(8967612500\) \([2]\) \(10240\) \(2.0400\)  
770.c2 770c4 \([1, -1, 0, -214949, -31495495]\) \(1098325674097093229481/205612182617187500\) \(205612182617187500\) \([4]\) \(10240\) \(2.0400\)  
770.c3 770c2 \([1, -1, 0, -204169, -35456067]\) \(941226862950447171561/45393906250000\) \(45393906250000\) \([2, 2]\) \(5120\) \(1.6934\)  
770.c4 770c1 \([1, -1, 0, -12089, -612755]\) \(-195395722614328041/50730248800000\) \(-50730248800000\) \([2]\) \(2560\) \(1.3469\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 770.c have rank \(0\).

Complex multiplication

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

Modular form 770.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - 3 q^{9} - q^{10} - q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} + 3 q^{18} + 4 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.