Properties

Label 333270.du
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("du1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.du1 333270du2 \([1, -1, 1, -74320103, -246561106913]\) \(420676324562824569/56350000000\) \(6081188489614350000000\) \([2]\) \(45416448\) \(3.1990\)  
333270.du2 333270du1 \([1, -1, 1, -4238183, -4554220769]\) \(-78013216986489/37918720000\) \(-4092118608782776320000\) \([2]\) \(22708224\) \(2.8524\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.du

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.