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Properties

Label 333270.m
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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

Elliptic curves in class 333270.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.m1 333270m3 \([1, -1, 0, -8050950, -8714672064]\) \(534774372149809/5323062500\) \(574455126965355562500\) \([2]\) \(21897216\) \(2.8020\)  
333270.m2 333270m4 \([1, -1, 0, -2099700, -21315848814]\) \(-9486391169809/1813439640250\) \(-195703074834049471610250\) \([2]\) \(43794432\) \(3.1485\)  
333270.m3 333270m1 \([1, -1, 0, -719010, 228028500]\) \(380920459249/12622400\) \(1362186221673614400\) \([2]\) \(7299072\) \(2.2527\) \(\Gamma_0(N)\)-optimal
333270.m4 333270m2 \([1, -1, 0, 233190, 787350780]\) \(12994449551/2489452840\) \(-268657177569578600040\) \([2]\) \(14598144\) \(2.5992\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.m have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.m do not have complex multiplication.

Modular form 333270.2.a.m

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} + 2 q^{13} + q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.