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Properties

Label 6378c
Number of curves $2$
Conductor $6378$
CM no
Rank $1$
Graph

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

Elliptic curves in class 6378c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6378.d1 6378c1 \([1, 0, 0, -1144386, 471132612]\) \(-165745346665991446425889/10662541623558144\) \(-10662541623558144\) \([7]\) \(98784\) \(2.1330\) \(\Gamma_0(N)\)-optimal
6378.d2 6378c2 \([1, 0, 0, 7869654, -13542161508]\) \(53900230693869615719525471/110424476261224735356024\) \(-110424476261224735356024\) \([]\) \(691488\) \(3.1059\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6378c have rank \(1\).

Complex multiplication

The elliptic curves in class 6378c do not have complex multiplication.

Modular form 6378.2.a.c

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2 q^{11} + q^{12} + q^{14} - q^{15} + q^{16} - 3 q^{17} + q^{18} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.