Properties

Label 71478i
Number of curves $2$
Conductor $71478$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 71478i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
71478.k2 71478i1 \([1, -1, 0, -3213870, -1637465896]\) \(296518892481/77948684\) \(965084017345524284076\) \([]\) \(3064320\) \(2.7353\) \(\Gamma_0(N)\)-optimal
71478.k1 71478i2 \([1, -1, 0, -2712587460, 54378741733712]\) \(178286568215258258721/180224\) \(2231356490150363136\) \([]\) \(21450240\) \(3.7083\)  

Rank

sage: E.rank()
 

The elliptic curves in class 71478i have rank \(0\).

Complex multiplication

The elliptic curves in class 71478i do not have complex multiplication.

Modular form 71478.2.a.i

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