# Properties

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

# Related objects

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)

$$q - q^{5} + q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.