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Properties

Label 444675cs
Number of curves $6$
Conductor $444675$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("cs1")
 
E.isogeny_class()
 

Elliptic curves in class 444675cs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
444675.cs6 444675cs1 \([1, 0, 0, 5184787, -1320002849208]\) \(4733169839/231139696095\) \(-752729462393250610194609375\) \([2]\) \(132710400\) \(3.8360\) \(\Gamma_0(N)\)-optimal*
444675.cs5 444675cs2 \([1, 0, 0, -1774256338, -28255403158333]\) \(189674274234120481/3859869269025\) \(12570049060665299131925390625\) \([2, 2]\) \(265420800\) \(4.1826\) \(\Gamma_0(N)\)-optimal*
444675.cs4 444675cs3 \([1, 0, 0, -3771588213, 47014048551042]\) \(1821931919215868881/761147600816295\) \(2478753040018191924292972734375\) \([2]\) \(530841600\) \(4.5292\) \(\Gamma_0(N)\)-optimal*
444675.cs2 444675cs4 \([1, 0, 0, -28247982463, -1827383356887208]\) \(765458482133960722801/326869475625\) \(1064483032628216731728515625\) \([2, 2]\) \(530841600\) \(4.5292\)  
444675.cs3 444675cs5 \([1, 0, 0, -28107909838, -1846402838127583]\) \(-754127868744065783521/15825714261328125\) \(-51538016139913099104620361328125\) \([2]\) \(1061683200\) \(4.8758\)  
444675.cs1 444675cs6 \([1, 0, 0, -451967673088, -116952447019390333]\) \(3135316978843283198764801/571725\) \(1861879457130992578125\) \([2]\) \(1061683200\) \(4.8758\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 444675cs1.

Rank

sage: E.rank()
 

The elliptic curves in class 444675cs have rank \(0\).

Complex multiplication

The elliptic curves in class 444675cs do not have complex multiplication.

Modular form 444675.2.a.cs

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.