↑

Properties

Label 68445.p
Number of curves $2$
Conductor $68445$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 68445.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68445.p1 68445w2 \([1, -1, 1, -38057, -2860386]\) \(-15590912409/78125\) \(-30544650703125\) \([]\) \(193536\) \(1.4348\)  
68445.p2 68445w1 \([1, -1, 1, -32, 2136]\) \(-9/5\) \(-1954857645\) \([]\) \(27648\) \(0.46188\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 68445.p have rank \(0\).

Complex multiplication

The elliptic curves in class 68445.p do not have complex multiplication.

Modular form 68445.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} + 3 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 3 q^{14} - q^{16} + 4 q^{17} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.