Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -300, -4048]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -300, -4048]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -300, -4048]) 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 4032l have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4032l do not have complex multiplication.

Modular form 4032.2.a.l

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);
 
\(q + q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

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

Elliptic curves in class 4032l

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
4032.w5 4032l1 \([0, 0, 0, -300, -4048]\) \(-15625/28\) \(-5350883328\) \([2]\) \(1536\) \(0.55694\) \(\Gamma_0(N)\)-optimal
4032.w4 4032l2 \([0, 0, 0, -6060, -181456]\) \(128787625/98\) \(18728091648\) \([2]\) \(3072\) \(0.90352\)  
4032.w6 4032l3 \([0, 0, 0, 2580, 84656]\) \(9938375/21952\) \(-4195092529152\) \([2]\) \(4608\) \(1.1062\)  
4032.w3 4032l4 \([0, 0, 0, -20460, 923312]\) \(4956477625/941192\) \(179864592187392\) \([2]\) \(9216\) \(1.4528\)  
4032.w2 4032l5 \([0, 0, 0, -98220, 11882288]\) \(-548347731625/1835008\) \(-350675489783808\) \([2]\) \(13824\) \(1.6556\)  
4032.w1 4032l6 \([0, 0, 0, -1572780, 759189296]\) \(2251439055699625/25088\) \(4794391461888\) \([2]\) \(27648\) \(2.0021\)