Properties

Label 215475.p
Number of curves $4$
Conductor $215475$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 215475.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
215475.p1 215475w4 \([1, 1, 1, -65935438, -206102914594]\) \(420339554066191969/244298925\) \(18424753904380078125\) \([2]\) \(16515072\) \(3.0214\)  
215475.p2 215475w2 \([1, 1, 1, -4144813, -3182502094]\) \(104413920565969/2472575625\) \(186478910623916015625\) \([2, 2]\) \(8257536\) \(2.6748\)  
215475.p3 215475w1 \([1, 1, 1, -574688, 94872656]\) \(278317173889/109245825\) \(8239198926912890625\) \([2]\) \(4128768\) \(2.3282\) \(\Gamma_0(N)\)-optimal
215475.p4 215475w3 \([1, 1, 1, 523812, -9942671094]\) \(210751100351/566398828125\) \(-42717171268487548828125\) \([2]\) \(16515072\) \(3.0214\)  

Rank

sage: E.rank()
 

The elliptic curves in class 215475.p have rank \(1\).

Complex multiplication

The elliptic curves in class 215475.p do not have complex multiplication.

Modular form 215475.2.a.p

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