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Properties

Label 12480bm
Number of curves $6$
Conductor $12480$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -37181, -2974275]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -37181, -2974275]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -37181, -2974275]) 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 12480bm have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 12480bm do not have complex multiplication.

Modular form 12480.2.a.bm

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);
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} - q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 12480bm

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
12480.m5 12480bm1 \([0, -1, 0, -37181, -2974275]\) \(-5551350318708736/550618236675\) \(-563833074355200\) \([2]\) \(61440\) \(1.5710\) \(\Gamma_0(N)\)-optimal
12480.m4 12480bm2 \([0, -1, 0, -608401, -182451599]\) \(1520107298839022416/13013105625\) \(213206722560000\) \([2, 2]\) \(122880\) \(1.9176\)  
12480.m1 12480bm3 \([0, -1, 0, -9734401, -11686687199]\) \(1556580279686303289604/114075\) \(7476019200\) \([2]\) \(245760\) \(2.2641\)  
12480.m3 12480bm4 \([0, -1, 0, -621921, -173904255]\) \(405929061432816484/35083409765625\) \(2299226342400000000\) \([2, 2]\) \(245760\) \(2.2641\)  
12480.m2 12480bm5 \([0, -1, 0, -2138241, 1005489441]\) \(8248670337458940482/1446075439453125\) \(189540000000000000000\) \([2]\) \(491520\) \(2.6107\)  
12480.m6 12480bm6 \([0, -1, 0, 678079, -806484255]\) \(263059523447441758/2294739983908125\) \(-300776159170805760000\) \([2]\) \(491520\) \(2.6107\)