Properties

Label 69828i
Number of curves $2$
Conductor $69828$
CM no
Rank $1$
Graph

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

Elliptic curves in class 69828i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69828.t2 69828i1 \([0, -1, 0, -5398621, 4829856658]\) \(7346581704933376/275517\) \(652582464473808\) \([2]\) \(1520640\) \(2.3351\) \(\Gamma_0(N)\)-optimal
69828.t1 69828i2 \([0, -1, 0, -5406556, 4814954728]\) \(461188987116496/2811467307\) \(106546703919662316288\) \([2]\) \(3041280\) \(2.6817\)  

Rank

sage: E.rank()
 

The elliptic curves in class 69828i have rank \(1\).

Complex multiplication

The elliptic curves in class 69828i do not have complex multiplication.

Modular form 69828.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{3} + 4 q^{5} + q^{9} - q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.