Properties

Label 56784.e
Number of curves $3$
Conductor $56784$
CM no
Rank $1$
Graph

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

Elliptic curves in class 56784.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
56784.e1 56784cc3 \([0, -1, 0, -70456832, 227658891264]\) \(-1956469094246217097/36641439744\) \(-724423602705600086016\) \([]\) \(7838208\) \(3.1271\)  
56784.e2 56784cc2 \([0, -1, 0, -328592, 694378944]\) \(-198461344537/10417365504\) \(-205957667106802630656\) \([]\) \(2612736\) \(2.5778\)  
56784.e3 56784cc1 \([0, -1, 0, 36448, -25479936]\) \(270840023/14329224\) \(-283297494492020736\) \([]\) \(870912\) \(2.0285\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 56784.e have rank \(1\).

Complex multiplication

The elliptic curves in class 56784.e do not have complex multiplication.

Modular form 56784.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3 q^{5} + q^{7} + q^{9} + 3 q^{11} + 3 q^{15} - 3 q^{17} - 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.