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Properties

Label 408135.r
Number of curves $4$
Conductor $408135$
CM no
Rank $0$
Graph

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

Elliptic curves in class 408135.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
408135.r1 408135r3 \([1, 0, 0, -725605, 237841772]\) \(8753151307882969/65205\) \(314732080845\) \([2]\) \(3244032\) \(1.8008\) \(\Gamma_0(N)\)-optimal*
408135.r2 408135r2 \([1, 0, 0, -45380, 3708327]\) \(2141202151369/5832225\) \(28151036120025\) \([2, 2]\) \(1622016\) \(1.4542\) \(\Gamma_0(N)\)-optimal*
408135.r3 408135r4 \([1, 0, 0, -27635, 6643350]\) \(-483551781049/3672913125\) \(-17728450127968125\) \([2]\) \(3244032\) \(1.8008\)  
408135.r4 408135r1 \([1, 0, 0, -3975, 6720]\) \(1439069689/828345\) \(3998263101105\) \([2]\) \(811008\) \(1.1077\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 408135.r1.

Rank

sage: E.rank()
 

The elliptic curves in class 408135.r have rank \(0\).

Complex multiplication

The elliptic curves in class 408135.r do not have complex multiplication.

Modular form 408135.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + q^{14} + q^{15} - q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.