Properties

Label 47040r
Number of curves $4$
Conductor $47040$
CM no
Rank $0$
Graph

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

Elliptic curves in class 47040r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.i3 47040r1 \([0, -1, 0, -44116, -3551834]\) \(1261112198464/675\) \(5082436800\) \([2]\) \(110592\) \(1.1920\) \(\Gamma_0(N)\)-optimal
47040.i2 47040r2 \([0, -1, 0, -44361, -3510135]\) \(20034997696/455625\) \(219561269760000\) \([2, 2]\) \(221184\) \(1.5386\)  
47040.i4 47040r3 \([0, -1, 0, 4639, -10889535]\) \(2863288/13286025\) \(-51219253009612800\) \([2]\) \(442368\) \(1.8852\)  
47040.i1 47040r4 \([0, -1, 0, -97281, 6534081]\) \(26410345352/10546875\) \(40659494400000000\) \([2]\) \(442368\) \(1.8852\)  

Rank

sage: E.rank()
 

The elliptic curves in class 47040r have rank \(0\).

Complex multiplication

The elliptic curves in class 47040r do not have complex multiplication.

Modular form 47040.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.