Properties

Label 452400m
Number of curves $4$
Conductor $452400$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 452400m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
452400.m3 452400m1 \([0, -1, 0, -709008, -229543488]\) \(615882348586441/21715200\) \(1389772800000000\) \([2]\) \(7077888\) \(1.9955\) \(\Gamma_0(N)\)-optimal*
452400.m2 452400m2 \([0, -1, 0, -741008, -207655488]\) \(703093388853961/115124490000\) \(7367967360000000000\) \([2, 2]\) \(14155776\) \(2.3420\) \(\Gamma_0(N)\)-optimal*
452400.m1 452400m3 \([0, -1, 0, -3341008, 2153144512]\) \(64443098670429961/6032611833300\) \(386087157331200000000\) \([2]\) \(28311552\) \(2.6886\) \(\Gamma_0(N)\)-optimal*
452400.m4 452400m4 \([0, -1, 0, 1346992, -1168135488]\) \(4223169036960119/11647532812500\) \(-745442100000000000000\) \([2]\) \(28311552\) \(2.6886\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 452400m1.

Rank

sage: E.rank()
 

The elliptic curves in class 452400m have rank \(1\).

Complex multiplication

The elliptic curves in class 452400m do not have complex multiplication.

Modular form 452400.2.a.m

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

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

The vertices are labelled with Cremona labels.