Properties

Label 77315.f
Number of curves $2$
Conductor $77315$
CM no
Rank $1$
Graph

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

Elliptic curves in class 77315.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77315.f1 77315d1 \([1, -1, 0, -11459, 135888]\) \(15438249/8225\) \(88659046081025\) \([2]\) \(194304\) \(1.3677\) \(\Gamma_0(N)\)-optimal
77315.f2 77315d2 \([1, -1, 0, 43766, 1030533]\) \(860085351/541205\) \(-5833765232131445\) \([2]\) \(388608\) \(1.7142\)  

Rank

sage: E.rank()
 

The elliptic curves in class 77315.f have rank \(1\).

Complex multiplication

The elliptic curves in class 77315.f do not have complex multiplication.

Modular form 77315.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - q^{7} - 3q^{8} - 3q^{9} + q^{10} - 6q^{11} - 2q^{13} - q^{14} - q^{16} - 4q^{17} - 3q^{18} - 4q^{19} + O(q^{20})\)  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.