Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 72450.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.r1 72450be4 \([1, -1, 0, -678298167, -18483604259]\) \(3029968325354577848895529/1753440696000000000000\) \(19972785427875000000000000000\) \([2]\) \(53084160\) \(4.1198\)  
72450.r2 72450be2 \([1, -1, 0, -466614792, -3879452132384]\) \(986396822567235411402169/6336721794060000\) \(72179221685464687500000\) \([2]\) \(17694720\) \(3.5705\)  
72450.r3 72450be1 \([1, -1, 0, -28602792, -63053576384]\) \(-227196402372228188089/19338934824115200\) \(-220282554480937200000000\) \([2]\) \(8847360\) \(3.2239\) \(\Gamma_0(N)\)-optimal
72450.r4 72450be3 \([1, -1, 0, 169573833, -2374036259]\) \(47342661265381757089751/27397579603968000000\) \(-312075555176448000000000000\) \([2]\) \(26542080\) \(3.7732\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 72450.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.