Properties

Label 75712.u
Number of curves $2$
Conductor $75712$
CM no
Rank $0$
Graph

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

Elliptic curves in class 75712.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.u1 75712df1 \([0, -1, 0, -178689, 30643105]\) \(-226981/14\) \(-38918682370899968\) \([]\) \(599040\) \(1.9378\) \(\Gamma_0(N)\)-optimal
75712.u2 75712df2 \([0, -1, 0, 524351, -1854769567]\) \(5735339/537824\) \(-1495100101960493170688\) \([]\) \(2995200\) \(2.7425\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75712.u have rank \(0\).

Complex multiplication

The elliptic curves in class 75712.u do not have complex multiplication.

Modular form 75712.2.a.u

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.