Properties

Label 42350.bl
Number of curves $6$
Conductor $42350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bl1")
 
E.isogeny_class()
 

Elliptic curves in class 42350.bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42350.bl1 42350bb6 \([1, 1, 0, -8259825, 9133575125]\) \(2251439055699625/25088\) \(694451912000000\) \([2]\) \(1244160\) \(2.4168\)  
42350.bl2 42350bb5 \([1, 1, 0, -515825, 142791125]\) \(-548347731625/1835008\) \(-50794196992000000\) \([2]\) \(622080\) \(2.0702\)  
42350.bl3 42350bb4 \([1, 1, 0, -107450, 11067500]\) \(4956477625/941192\) \(26052797511125000\) \([2]\) \(414720\) \(1.8675\)  
42350.bl4 42350bb2 \([1, 1, 0, -31825, -2197125]\) \(128787625/98\) \(2712702781250\) \([2]\) \(138240\) \(1.3182\)  
42350.bl5 42350bb1 \([1, 1, 0, -1575, -49375]\) \(-15625/28\) \(-775057937500\) \([2]\) \(69120\) \(0.97158\) \(\Gamma_0(N)\)-optimal
42350.bl6 42350bb3 \([1, 1, 0, 13550, 1024500]\) \(9938375/21952\) \(-607645423000000\) \([2]\) \(207360\) \(1.5209\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42350.bl have rank \(1\).

Complex multiplication

The elliptic curves in class 42350.bl do not have complex multiplication.

Modular form 42350.2.a.bl

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{12} - 4 q^{13} - q^{14} + q^{16} + 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.