Properties

Label 53312.cc
Number of curves $4$
Conductor $53312$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cc1")
 
E.isogeny_class()
 

Elliptic curves in class 53312.cc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53312.cc1 53312cb4 \([0, -1, 0, -354433, -56005599]\) \(159661140625/48275138\) \(1488852539293564928\) \([2]\) \(663552\) \(2.1922\)  
53312.cc2 53312cb3 \([0, -1, 0, -323073, -70562911]\) \(120920208625/19652\) \(606086928269312\) \([2]\) \(331776\) \(1.8456\)  
53312.cc3 53312cb2 \([0, -1, 0, -134913, 19114145]\) \(8805624625/2312\) \(71304344502272\) \([2]\) \(221184\) \(1.6429\)  
53312.cc4 53312cb1 \([0, -1, 0, -9473, 222881]\) \(3048625/1088\) \(33554985648128\) \([2]\) \(110592\) \(1.2963\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 53312.cc have rank \(0\).

Complex multiplication

The elliptic curves in class 53312.cc do not have complex multiplication.

Modular form 53312.2.a.cc

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} + q^{9} + 6 q^{11} + 2 q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.