Properties

Label 388416.v
Number of curves $6$
Conductor $388416$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("v1")
 
E.isogeny_class()
 

Elliptic curves in class 388416.v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.v1 388416v5 \([0, -1, 0, -14501249, 21259595553]\) \(53297461115137/147\) \(930145276526592\) \([2]\) \(9437184\) \(2.5305\)  
388416.v2 388416v3 \([0, -1, 0, -906689, 332129889]\) \(13027640977/21609\) \(136731355649409024\) \([2, 2]\) \(4718592\) \(2.1840\)  
388416.v3 388416v4 \([0, -1, 0, -721729, -234328607]\) \(6570725617/45927\) \(290603959966236672\) \([2]\) \(4718592\) \(2.1840\)  
388416.v4 388416v6 \([0, -1, 0, -629249, 538933665]\) \(-4354703137/17294403\) \(-109430661638077022208\) \([2]\) \(9437184\) \(2.5305\)  
388416.v5 388416v2 \([0, -1, 0, -74369, 1698849]\) \(7189057/3969\) \(25113922466217984\) \([2, 2]\) \(2359296\) \(1.8374\)  
388416.v6 388416v1 \([0, -1, 0, 18111, 200673]\) \(103823/63\) \(-398633689939968\) \([2]\) \(1179648\) \(1.4908\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 388416.v have rank \(2\).

Complex multiplication

The elliptic curves in class 388416.v do not have complex multiplication.

Modular form 388416.2.a.v

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.