Properties

Label 72450.b
Number of curves $4$
Conductor $72450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 72450.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.b1 72450bk4 \([1, -1, 0, -341210367, 2414269686541]\) \(385693937170561837203625/2159357734550274048\) \(24596434195111715328000000\) \([2]\) \(33177600\) \(3.7144\)  
72450.b2 72450bk2 \([1, -1, 0, -25198992, -46382343584]\) \(155355156733986861625/8291568305839392\) \(94446145233701824500000\) \([2]\) \(11059200\) \(3.1651\)  
72450.b3 72450bk3 \([1, -1, 0, -9434367, 79561974541]\) \(-8152944444844179625/235342826399858688\) \(-2680701881960890368000000\) \([2]\) \(16588800\) \(3.3678\)  
72450.b4 72450bk1 \([1, -1, 0, 1045008, -2896035584]\) \(11079872671250375/324440155855872\) \(-3695576150295792000000\) \([2]\) \(5529600\) \(2.8185\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 72450.b have rank \(1\).

Complex multiplication

The elliptic curves in class 72450.b do not have complex multiplication.

Modular form 72450.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} - 6q^{11} - 2q^{13} + q^{14} + q^{16} - 6q^{17} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.