Properties

Label 3450.j
Number of curves $4$
Conductor $3450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3450.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3450.j1 3450j3 \([1, 0, 1, -34560276, -78204168302]\) \(292169767125103365085489/72534787200\) \(1133356050000000\) \([2]\) \(172032\) \(2.7043\)  
3450.j2 3450j4 \([1, 0, 1, -2528276, -777224302]\) \(114387056741228939569/49503729150000000\) \(773495767968750000000\) \([2]\) \(172032\) \(2.7043\)  
3450.j3 3450j2 \([1, 0, 1, -2160276, -1221768302]\) \(71356102305927901489/35540674560000\) \(555323040000000000\) \([2, 2]\) \(86016\) \(2.3578\)  
3450.j4 3450j1 \([1, 0, 1, -112276, -25736302]\) \(-10017490085065009/12502381363200\) \(-195349708800000000\) \([2]\) \(43008\) \(2.0112\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3450.j have rank \(1\).

Complex multiplication

The elliptic curves in class 3450.j do not have complex multiplication.

Modular form 3450.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + q^{12} - 6q^{13} + q^{16} - 2q^{17} - q^{18} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.