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Properties

Label 443760.j
Number of curves $8$
Conductor $443760$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 443760.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
443760.j1 443760j8 \([0, -1, 0, -63902056, -196595381840]\) \(1114544804970241/405\) \(10486382734725120\) \([2]\) \(20643840\) \(2.8646\)  
443760.j2 443760j6 \([0, -1, 0, -3994456, -3069870800]\) \(272223782641/164025\) \(4246985007563673600\) \([2, 2]\) \(10321920\) \(2.5180\)  
443760.j3 443760j7 \([0, -1, 0, -3254856, -4242580560]\) \(-147281603041/215233605\) \(-5572893726925052497920\) \([2]\) \(20643840\) \(2.8646\)  
443760.j4 443760j3 \([0, -1, 0, -2367336, 1402756656]\) \(56667352321/15\) \(388384545730560\) \([2]\) \(5160960\) \(2.1715\) \(\Gamma_0(N)\)-optimal*
443760.j5 443760j4 \([0, -1, 0, -296456, -28635600]\) \(111284641/50625\) \(1310797841840640000\) \([2, 2]\) \(5160960\) \(2.1715\)  
443760.j6 443760j2 \([0, -1, 0, -148536, 21775536]\) \(13997521/225\) \(5825768185958400\) \([2, 2]\) \(2580480\) \(1.8249\) \(\Gamma_0(N)\)-optimal*
443760.j7 443760j1 \([0, -1, 0, -616, 948400]\) \(-1/15\) \(-388384545730560\) \([2]\) \(1290240\) \(1.4783\) \(\Gamma_0(N)\)-optimal*
443760.j8 443760j5 \([0, -1, 0, 1034824, -216079824]\) \(4733169839/3515625\) \(-91027627905600000000\) \([2]\) \(10321920\) \(2.5180\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 443760.j1.

Rank

sage: E.rank()
 

The elliptic curves in class 443760.j have rank \(1\).

Complex multiplication

The elliptic curves in class 443760.j do not have complex multiplication.

Modular form 443760.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.