Properties

Label 27720p
Number of curves $4$
Conductor $27720$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27720p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27720.y4 27720p1 \([0, 0, 0, 3273, -69046]\) \(20777545136/23059575\) \(-4303470124800\) \([2]\) \(32768\) \(1.1107\) \(\Gamma_0(N)\)-optimal
27720.y3 27720p2 \([0, 0, 0, -18507, -648394]\) \(939083699236/300155625\) \(224064973440000\) \([2, 2]\) \(65536\) \(1.4573\)  
27720.y2 27720p3 \([0, 0, 0, -117507, 15013406]\) \(120186986927618/4332064275\) \(6467737306060800\) \([2]\) \(131072\) \(1.8039\)  
27720.y1 27720p4 \([0, 0, 0, -267987, -53388466]\) \(1425631925916578/270703125\) \(404157600000000\) \([2]\) \(131072\) \(1.8039\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27720p have rank \(1\).

Complex multiplication

The elliptic curves in class 27720p do not have complex multiplication.

Modular form 27720.2.a.p

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} - q^{11} - 2 q^{13} + 2 q^{17} + O(q^{20})\)  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 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.