Properties

Label 97020.z
Number of curves $4$
Conductor $97020$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 97020.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97020.z1 97020bu4 \([0, 0, 0, -21871983, 20758643318]\) \(52702650535889104/22020583921875\) \(483486736674351564000000\) \([2]\) \(11943936\) \(3.2417\)  
97020.z2 97020bu2 \([0, 0, 0, -18855543, 31514220542]\) \(33766427105425744/9823275\) \(215681073220166400\) \([2]\) \(3981312\) \(2.6924\)  
97020.z3 97020bu1 \([0, 0, 0, -1173648, 496640333]\) \(-130287139815424/2250652635\) \(-3088475939558061360\) \([2]\) \(1990656\) \(2.3458\) \(\Gamma_0(N)\)-optimal
97020.z4 97020bu3 \([0, 0, 0, 4541712, 2379994337]\) \(7549996227362816/6152409907875\) \(-8442693321606497646000\) \([2]\) \(5971968\) \(2.8951\)  

Rank

sage: E.rank()
 

The elliptic curves in class 97020.z have rank \(1\).

Complex multiplication

The elliptic curves in class 97020.z do not have complex multiplication.

Modular form 97020.2.a.z

sage: E.q_eigenform(10)
 
\(q - q^{5} + q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.