Properties

Label 12480cj
Number of curves $8$
Conductor $12480$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cj1")
 
E.isogeny_class()
 

Elliptic curves in class 12480cj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12480.ca6 12480cj1 \([0, 1, 0, -7041, -229761]\) \(147281603041/5265\) \(1380188160\) \([2]\) \(12288\) \(0.84284\) \(\Gamma_0(N)\)-optimal
12480.ca5 12480cj2 \([0, 1, 0, -7361, -208065]\) \(168288035761/27720225\) \(7266690662400\) \([2, 2]\) \(24576\) \(1.1894\)  
12480.ca4 12480cj3 \([0, 1, 0, -33281, 2129919]\) \(15551989015681/1445900625\) \(379034173440000\) \([2, 2]\) \(49152\) \(1.5360\)  
12480.ca7 12480cj4 \([0, 1, 0, 13439, -1152385]\) \(1023887723039/2798036865\) \(-733488575938560\) \([2]\) \(49152\) \(1.5360\)  
12480.ca2 12480cj5 \([0, 1, 0, -520001, 144154815]\) \(59319456301170001/594140625\) \(155750400000000\) \([2, 2]\) \(98304\) \(1.8826\)  
12480.ca8 12480cj6 \([0, 1, 0, 38719, 10150719]\) \(24487529386319/183539412225\) \(-48113755678310400\) \([2]\) \(98304\) \(1.8826\)  
12480.ca1 12480cj7 \([0, 1, 0, -8320001, 9234274815]\) \(242970740812818720001/24375\) \(6389760000\) \([2]\) \(196608\) \(2.2291\)  
12480.ca3 12480cj8 \([0, 1, 0, -507521, 151415679]\) \(-55150149867714721/5950927734375\) \(-1560000000000000000\) \([2]\) \(196608\) \(2.2291\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12480cj have rank \(1\).

Complex multiplication

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

Modular form 12480.2.a.cj

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} + 4 q^{11} - q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.