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Properties

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

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -8259825, 9133575125]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -8259825, 9133575125]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -8259825, 9133575125]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

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

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 42350.2.a.bl

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{12} - 4 q^{13} - q^{14} + q^{16} + 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 42350.bl

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
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 \([1, 1, 0, 13550, 1024500]\) \(9938375/21952\) \(-607645423000000\) \([2]\) \(207360\) \(1.5209\)