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Properties

Label 482790.cc
Number of curves $8$
Conductor $482790$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cc1")
 
E.isogeny_class()
 

Elliptic curves in class 482790.cc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
482790.cc1 482790cc7 \([1, 0, 1, -18223586879, 946886902102502]\) \(377806291534052689568887263169/100912963819335937500\) \(178773471096746592773437500\) \([2]\) \(716636160\) \(4.4089\) \(\Gamma_0(N)\)-optimal*
482790.cc2 482790cc8 \([1, 0, 1, -2284774759, -19494281336554]\) \(744556702832013561199553089/338208906180283330846500\) \(599157708041648917877756386500\) \([2]\) \(716636160\) \(4.4089\)  
482790.cc3 482790cc5 \([1, 0, 1, -1924686019, -32500490840578]\) \(445089424735238304524848129/206488340640267840\) \(365806691233013534898240\) \([2]\) \(238878720\) \(3.8595\)  
482790.cc4 482790cc6 \([1, 0, 1, -1143442259, 14673104916446]\) \(93327647066813251630073089/1506876757438610250000\) \(2669524095284701813100250000\) \([2, 2]\) \(358318080\) \(4.0623\) \(\Gamma_0(N)\)-optimal*
482790.cc5 482790cc4 \([1, 0, 1, -259958339, 868268524286]\) \(1096677312076899338462209/450803852032204440000\) \(798626522910024129930840000\) \([2]\) \(238878720\) \(3.8595\) \(\Gamma_0(N)\)-optimal*
482790.cc6 482790cc2 \([1, 0, 1, -120914819, -502311261058]\) \(110358600993178429667329/2339305154932838400\) \(4144221779577974128742400\) \([2, 2]\) \(119439360\) \(3.5130\) \(\Gamma_0(N)\)-optimal*
482790.cc7 482790cc3 \([1, 0, 1, -4599939, 641200923262]\) \(-6076082794014148609/100253882690711904000\) \(-177605868673440271362144000\) \([2]\) \(179159040\) \(3.7157\) \(\Gamma_0(N)\)-optimal*
482790.cc8 482790cc1 \([1, 0, 1, 511101, -23747425154]\) \(8334681620170751/137523678664458240\) \(-243631585698486304112640\) \([2]\) \(59719680\) \(3.1664\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 482790.cc1.

Rank

sage: E.rank()
 

The elliptic curves in class 482790.cc have rank \(1\).

Complex multiplication

The elliptic curves in class 482790.cc do not have complex multiplication.

Modular form 482790.2.a.cc

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} - q^{18} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 4 & 12 \\ 4 & 1 & 3 & 2 & 12 & 6 & 4 & 12 \\ 12 & 3 & 1 & 6 & 4 & 2 & 12 & 4 \\ 2 & 2 & 6 & 1 & 6 & 3 & 2 & 6 \\ 3 & 12 & 4 & 6 & 1 & 2 & 12 & 4 \\ 6 & 6 & 2 & 3 & 2 & 1 & 6 & 2 \\ 4 & 4 & 12 & 2 & 12 & 6 & 1 & 3 \\ 12 & 12 & 4 & 6 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.