Properties

Label 369600.mn
Number of curves $4$
Conductor $369600$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("mn1")
 
E.isogeny_class()
 

Elliptic curves in class 369600.mn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.mn1 369600mn4 \([0, 1, 0, -1409633, -644647137]\) \(75627935783569/396165\) \(1622691840000000\) \([2]\) \(4718592\) \(2.1146\)  
369600.mn2 369600mn2 \([0, 1, 0, -89633, -9727137]\) \(19443408769/1334025\) \(5464166400000000\) \([2, 2]\) \(2359296\) \(1.7680\)  
369600.mn3 369600mn1 \([0, 1, 0, -17633, 712863]\) \(148035889/31185\) \(127733760000000\) \([2]\) \(1179648\) \(1.4214\) \(\Gamma_0(N)\)-optimal
369600.mn4 369600mn3 \([0, 1, 0, 78367, -41815137]\) \(12994449551/192163125\) \(-787100160000000000\) \([2]\) \(4718592\) \(2.1146\)  

Rank

sage: E.rank()
 

The elliptic curves in class 369600.mn have rank \(1\).

Complex multiplication

The elliptic curves in class 369600.mn do not have complex multiplication.

Modular form 369600.2.a.mn

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} - q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content 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.