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Properties

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

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 77315.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77315.e1 77315c2 \([1, -1, 0, -1730, -27249]\) \(5517084663/4375\) \(454225625\) \([2]\) \(45312\) \(0.59161\)  
77315.e2 77315c1 \([1, -1, 0, -85, -600]\) \(-658503/1225\) \(-127183175\) \([2]\) \(22656\) \(0.24504\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 77315.2.a.e

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