Properties

Label 77077l
Number of curves $4$
Conductor $77077$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 77077l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77077.u4 77077l1 \([1, -1, 0, -93011, -92039480]\) \(-426957777/17320303\) \(-3609938775122646967\) \([2]\) \(875520\) \(2.2410\) \(\Gamma_0(N)\)-optimal
77077.u3 77077l2 \([1, -1, 0, -3680056, -2701256013]\) \(26444947540257/169338169\) \(35293864222893317041\) \([2, 2]\) \(1751040\) \(2.5876\)  
77077.u2 77077l3 \([1, -1, 0, -5962721, 1051901780]\) \(112489728522417/62811265517\) \(13091273455455273091013\) \([2]\) \(3502080\) \(2.9341\)  
77077.u1 77077l4 \([1, -1, 0, -58790111, -173487316458]\) \(107818231938348177/4463459\) \(930284748209667851\) \([2]\) \(3502080\) \(2.9341\)  

Rank

sage: E.rank()
 

The elliptic curves in class 77077l have rank \(1\).

Complex multiplication

The elliptic curves in class 77077l do not have complex multiplication.

Modular form 77077.2.a.l

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.