Properties

Label 425.d
Number of curves $4$
Conductor $425$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 425.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
425.d1 425a3 \([1, -1, 0, -2267, -40984]\) \(82483294977/17\) \(265625\) \([2]\) \(128\) \(0.42808\)  
425.d2 425a2 \([1, -1, 0, -142, -609]\) \(20346417/289\) \(4515625\) \([2, 2]\) \(64\) \(0.081509\)  
425.d3 425a4 \([1, -1, 0, -17, -1734]\) \(-35937/83521\) \(-1305015625\) \([2]\) \(128\) \(0.42808\)  
425.d4 425a1 \([1, -1, 0, -17, 16]\) \(35937/17\) \(265625\) \([2]\) \(32\) \(-0.26506\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 425.d have rank \(1\).

Complex multiplication

The elliptic curves in class 425.d do not have complex multiplication.

Modular form 425.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8} - 3 q^{9} + 2 q^{13} - 4 q^{14} - q^{16} - q^{17} - 3 q^{18} - 4 q^{19} + O(q^{20})\) 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.