Properties

Label 425880i
Number of curves $4$
Conductor $425880$
CM no
Rank $1$
Graph

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

Elliptic curves in class 425880i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
425880.i4 425880i1 \([0, 0, 0, 13182, -173563]\) \(4499456/2835\) \(-159610216998960\) \([2]\) \(1179648\) \(1.4143\) \(\Gamma_0(N)\)-optimal*
425880.i3 425880i2 \([0, 0, 0, -55263, -1419262]\) \(20720464/11025\) \(9931302391046400\) \([2, 2]\) \(2359296\) \(1.7609\) \(\Gamma_0(N)\)-optimal*
425880.i2 425880i3 \([0, 0, 0, -511563, 139759958]\) \(4108974916/36015\) \(129769017909672960\) \([2]\) \(4718592\) \(2.1075\) \(\Gamma_0(N)\)-optimal*
425880.i1 425880i4 \([0, 0, 0, -694083, -222323218]\) \(10262905636/13125\) \(47291916147840000\) \([2]\) \(4718592\) \(2.1075\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 425880i1.

Rank

sage: E.rank()
 

The elliptic curves in class 425880i have rank \(1\).

Complex multiplication

The elliptic curves in class 425880i do not have complex multiplication.

Modular form 425880.2.a.i

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.