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Properties

Label 450450lr
Number of curves $4$
Conductor $450450$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("lr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 450450lr have rank \(0\).

Complex multiplication

The elliptic curves in class 450450lr do not have complex multiplication.

Modular form 450450.2.a.lr

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

Isogeny matrix

Copy content 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 Cremona 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 450450lr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450450.lr3 450450lr1 \([1, -1, 1, -117605, 15552397]\) \(15792469779969/400400\) \(4560806250000\) \([2]\) \(2752512\) \(1.5368\) \(\Gamma_0(N)\)-optimal
450450.lr2 450450lr2 \([1, -1, 1, -122105, 14301397]\) \(17675559395649/2505002500\) \(28533544101562500\) \([2, 2]\) \(5505024\) \(1.8834\)  
450450.lr4 450450lr3 \([1, -1, 1, 199645, 76720897]\) \(77259787831071/268236718750\) \(-3055383874511718750\) \([2]\) \(11010048\) \(2.2300\)  
450450.lr1 450450lr4 \([1, -1, 1, -515855, -128236103]\) \(1332779492447649/146356560350\) \(1667092695236718750\) \([2]\) \(11010048\) \(2.2300\)