Properties

Label 37570k
Number of curves $4$
Conductor $37570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 37570k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37570.l3 37570k1 \([1, -1, 1, -457108, 117956727]\) \(437608510454961/4707123200\) \(113618511031500800\) \([4]\) \(589824\) \(2.0882\) \(\Gamma_0(N)\)-optimal
37570.l2 37570k2 \([1, -1, 1, -827028, -99852169]\) \(2591733435976881/1320660640000\) \(31877537323584160000\) \([2, 2]\) \(1179648\) \(2.4347\)  
37570.l4 37570k3 \([1, -1, 1, 3080252, -775030153]\) \(133902615693854799/88219056250000\) \(-2129393557349256250000\) \([2]\) \(2359296\) \(2.7813\)  
37570.l1 37570k4 \([1, -1, 1, -10653028, -13368882569]\) \(5539229398623592881/5546968902800\) \(133890344632189293200\) \([2]\) \(2359296\) \(2.7813\)  

Rank

sage: E.rank()
 

The elliptic curves in class 37570k have rank \(1\).

Complex multiplication

The elliptic curves in class 37570k do not have complex multiplication.

Modular form 37570.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.