# Properties

 Label 97020.w Number of curves $2$ Conductor $97020$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 97020.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97020.w1 97020c1 $$[0, 0, 0, -110103, 14170702]$$ $$-3704530032/33275$$ $$-1325885782636800$$ $$$$ $$580608$$ $$1.7250$$ $$\Gamma_0(N)$$-optimal
97020.w2 97020c2 $$[0, 0, 0, 342657, 74810358]$$ $$153174672/171875$$ $$-4992617435652000000$$ $$[]$$ $$1741824$$ $$2.2743$$

## Rank

sage: E.rank()

The elliptic curves in class 97020.w have rank $$1$$.

## Complex multiplication

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

## Modular form 97020.2.a.w

sage: E.q_eigenform(10)

$$q - q^{5} + q^{11} - 4q^{13} - 3q^{17} + 5q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 