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Properties

Label 75810.l
Number of curves $8$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 75810.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75810.l1 75810l8 \([1, 1, 0, -2328818, -861559362]\) \(29689921233686449/10380965400750\) \(488381662908801810750\) \([2]\) \(3981312\) \(2.6712\)  
75810.l2 75810l5 \([1, 1, 0, -2079728, -1155271128]\) \(21145699168383889/2593080\) \(121993733103480\) \([2]\) \(1327104\) \(2.1219\)  
75810.l3 75810l6 \([1, 1, 0, -975068, 360335388]\) \(2179252305146449/66177562500\) \(3113381730245062500\) \([2, 2]\) \(1990656\) \(2.3247\)  
75810.l4 75810l3 \([1, 1, 0, -967848, 366083952]\) \(2131200347946769/2058000\) \(96820423098000\) \([2]\) \(995328\) \(1.9781\)  
75810.l5 75810l2 \([1, 1, 0, -130328, -17991168]\) \(5203798902289/57153600\) \(2688841464321600\) \([2, 2]\) \(663552\) \(1.7754\)  
75810.l6 75810l4 \([1, 1, 0, -29248, -45060392]\) \(-58818484369/18600435000\) \(-875073851558235000\) \([2]\) \(1327104\) \(2.1219\)  
75810.l7 75810l1 \([1, 1, 0, -14808, 237888]\) \(7633736209/3870720\) \(182101432504320\) \([2]\) \(331776\) \(1.4288\) \(\Gamma_0(N)\)-optimal
75810.l8 75810l7 \([1, 1, 0, 263162, 1214466442]\) \(42841933504271/13565917968750\) \(-638220562413574218750\) \([2]\) \(3981312\) \(2.6712\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75810.l have rank \(1\).

Complex multiplication

The elliptic curves in class 75810.l do not have complex multiplication.

Modular form 75810.2.a.l

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.