Properties

Label 162624.t
Number of curves $4$
Conductor $162624$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 162624.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162624.t1 162624cm3 \([0, -1, 0, -712483009, 7320216607585]\) \(86129359107301290313/9166294368\) \(4256864565164362432512\) \([2]\) \(44236800\) \(3.5780\)  
162624.t2 162624cm2 \([0, -1, 0, -44640449, 113794679649]\) \(21184262604460873/216872764416\) \(100716598106934051078144\) \([2, 2]\) \(22118400\) \(3.2314\)  
162624.t3 162624cm4 \([0, -1, 0, -11186369, 280362543969]\) \(-333345918055753/72923718045024\) \(-33866072683593272515362816\) \([2]\) \(44236800\) \(3.5780\)  
162624.t4 162624cm1 \([0, -1, 0, -4991169, -1418198175]\) \(29609739866953/15259926528\) \(7086772243626775805952\) \([2]\) \(11059200\) \(2.8848\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 162624.t have rank \(1\).

Complex multiplication

The elliptic curves in class 162624.t do not have complex multiplication.

Modular form 162624.2.a.t

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} + 2 q^{13} + 2 q^{15} - 6 q^{17} + 8 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.