Properties

Label 81312.o
Number of curves $2$
Conductor $81312$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 81312.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
81312.o1 81312bd2 \([0, -1, 0, -4033, -91727]\) \(1000000/63\) \(457147772928\) \([2]\) \(89600\) \(0.98840\)  
81312.o2 81312bd1 \([0, -1, 0, 202, -6180]\) \(8000/147\) \(-16666845888\) \([2]\) \(44800\) \(0.64183\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 81312.o have rank \(0\).

Complex multiplication

The elliptic curves in class 81312.o do not have complex multiplication.

Modular form 81312.2.a.o

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.