Properties

Label 58870.t
Number of curves $4$
Conductor $58870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 58870.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58870.t1 58870r4 \([1, -1, 1, -642682, 198125421]\) \(49354130009241/99019340\) \(58899012662028140\) \([2]\) \(645120\) \(2.1051\)  
58870.t2 58870r3 \([1, -1, 1, -541762, -152564851]\) \(29563822919961/174072500\) \(103542382544772500\) \([2]\) \(645120\) \(2.1051\)  
58870.t3 58870r2 \([1, -1, 1, -53982, 793181]\) \(29246580441/16483600\) \(9804829694035600\) \([2, 2]\) \(322560\) \(1.7585\)  
58870.t4 58870r1 \([1, -1, 1, 13298, 93469]\) \(437245479/259840\) \(-154558891728640\) \([4]\) \(161280\) \(1.4120\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 58870.t have rank \(1\).

Complex multiplication

The elliptic curves in class 58870.t do not have complex multiplication.

Modular form 58870.2.a.t

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.