Properties

Label 9075.a
Number of curves $2$
Conductor $9075$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 9075.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9075.a1 9075j1 \([0, -1, 1, -1008, 12968]\) \(-102400/3\) \(-3321676875\) \([]\) \(8400\) \(0.60585\) \(\Gamma_0(N)\)-optimal
9075.a2 9075j2 \([0, -1, 1, 5042, -610182]\) \(20480/243\) \(-168159891796875\) \([]\) \(42000\) \(1.4106\)  

Rank

sage: E.rank()
 

The elliptic curves in class 9075.a have rank \(1\).

Complex multiplication

The elliptic curves in class 9075.a do not have complex multiplication.

Modular form 9075.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + q^{9} - 2 q^{12} - q^{13} - 6 q^{14} - 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.