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Properties

Label 42978.u
Number of curves $2$
Conductor $42978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 42978.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.u1 42978r1 \([1, 0, 0, -459769306, 3796241996132]\) \(-10748395438529140294639078020769/5737242625602477531070464\) \(-5737242625602477531070464\) \([7]\) \(11063808\) \(3.7003\) \(\Gamma_0(N)\)-optimal
42978.u2 42978r2 \([1, 0, 0, 2127889334, -194909741524108]\) \(1065542619208351347902742829533791/17028260093251190608019507801424\) \(-17028260093251190608019507801424\) \([]\) \(77446656\) \(4.6733\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42978.u have rank \(1\).

Complex multiplication

The elliptic curves in class 42978.u do not have complex multiplication.

Modular form 42978.2.a.u

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2 q^{11} + q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.