Properties

Label 34225.e
Number of curves $3$
Conductor $34225$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 34225.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34225.e1 34225b3 \([0, -1, 1, -64114833, -197578150557]\) \(727057727488000/37\) \(1483310580203125\) \([]\) \(1181952\) \(2.8323\)  
34225.e2 34225b2 \([0, -1, 1, -798583, -265720682]\) \(1404928000/50653\) \(2030652184298078125\) \([]\) \(393984\) \(2.2830\)  
34225.e3 34225b1 \([0, -1, 1, -114083, 14753193]\) \(4096000/37\) \(1483310580203125\) \([]\) \(131328\) \(1.7336\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 34225.e have rank \(1\).

Complex multiplication

The elliptic curves in class 34225.e do not have complex multiplication.

Modular form 34225.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} + q^{7} - 2 q^{9} + 3 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{16} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.