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Properties

Label 15870.u
Number of curves $8$
Conductor $15870$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 15870.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15870.u1 15870u8 \([1, 0, 1, -2821433, 1823881556]\) \(16778985534208729/81000\) \(11990907009000\) \([2]\) \(304128\) \(2.1319\)  
15870.u2 15870u7 \([1, 0, 1, -239913, 6135988]\) \(10316097499609/5859375000\) \(867397787109375000\) \([2]\) \(304128\) \(2.1319\)  
15870.u3 15870u6 \([1, 0, 1, -176433, 28455556]\) \(4102915888729/9000000\) \(1332323001000000\) \([2, 2]\) \(152064\) \(1.7853\)  
15870.u4 15870u4 \([1, 0, 1, -152628, -22963244]\) \(2656166199049/33750\) \(4996211253750\) \([2]\) \(101376\) \(1.5826\)  
15870.u5 15870u5 \([1, 0, 1, -36248, 2284868]\) \(35578826569/5314410\) \(786723408860490\) \([2]\) \(101376\) \(1.5826\)  
15870.u6 15870u2 \([1, 0, 1, -9798, -338972]\) \(702595369/72900\) \(10791816308100\) \([2, 2]\) \(50688\) \(1.2360\)  
15870.u7 15870u3 \([1, 0, 1, -7153, 761348]\) \(-273359449/1536000\) \(-227383125504000\) \([2]\) \(76032\) \(1.4388\)  
15870.u8 15870u1 \([1, 0, 1, 782, -25804]\) \(357911/2160\) \(-319757520240\) \([2]\) \(25344\) \(0.88946\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 15870.u have rank \(0\).

Complex multiplication

The elliptic curves in class 15870.u do not have complex multiplication.

Modular form 15870.2.a.u

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.