Properties

Label 121680cz
Number of curves $4$
Conductor $121680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cz1")
 
E.isogeny_class()
 

Elliptic curves in class 121680cz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121680.db4 121680cz1 \([0, 0, 0, 263133, 40552226]\) \(3774555693/3515200\) \(-1876436471454105600\) \([2]\) \(2322432\) \(2.1936\) \(\Gamma_0(N)\)-optimal
121680.db3 121680cz2 \([0, 0, 0, -1359267, 365356706]\) \(520300455507/193072360\) \(103063273194616750080\) \([2]\) \(4644864\) \(2.5401\)  
121680.db2 121680cz3 \([0, 0, 0, -6064227, 5811245154]\) \(-63378025803/812500\) \(-316180239388416000000\) \([2]\) \(6967296\) \(2.7429\)  
121680.db1 121680cz4 \([0, 0, 0, -97324227, 369555353154]\) \(261984288445803/42250\) \(16441372448197632000\) \([2]\) \(13934592\) \(3.0894\)  

Rank

sage: E.rank()
 

The elliptic curves in class 121680cz have rank \(1\).

Complex multiplication

The elliptic curves in class 121680cz do not have complex multiplication.

Modular form 121680.2.a.cz

sage: E.q_eigenform(10)
 
\(q + q^{5} - 4 q^{7} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.