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Properties

Label 136367f
Number of curves $2$
Conductor $136367$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 136367f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
136367.f2 136367f1 \([1, -1, 1, -885520978, 10142786988744]\) \(-1074191725926252207/4097152081\) \(-292900798589856425116687\) \([2]\) \(67092480\) \(3.7170\) \(\Gamma_0(N)\)-optimal
136367.f1 136367f2 \([1, -1, 1, -14168348613, 649126493198054]\) \(4399901392374538640127/64009\) \(4575931487601062743\) \([2]\) \(134184960\) \(4.0636\)  

Rank

sage: E.rank()
 

The elliptic curves in class 136367f have rank \(0\).

Complex multiplication

The elliptic curves in class 136367f do not have complex multiplication.

Modular form 136367.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.