# Properties

 Label 180.a Number of curves $4$ Conductor $180$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 180.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
180.a1 180a3 $$[0, 0, 0, -372, 2761]$$ $$488095744/125$$ $$1458000$$ $$$$ $$36$$ $$0.16866$$
180.a2 180a4 $$[0, 0, 0, -327, 3454]$$ $$-20720464/15625$$ $$-2916000000$$ $$$$ $$72$$ $$0.51524$$
180.a3 180a1 $$[0, 0, 0, -12, -11]$$ $$16384/5$$ $$58320$$ $$$$ $$12$$ $$-0.38064$$ $$\Gamma_0(N)$$-optimal
180.a4 180a2 $$[0, 0, 0, 33, -74]$$ $$21296/25$$ $$-4665600$$ $$$$ $$24$$ $$-0.034070$$

## Rank

sage: E.rank()

The elliptic curves in class 180.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 180.a do not have complex multiplication.

## Modular form180.2.a.a

sage: E.q_eigenform(10)

$$q + q^{5} + 2q^{7} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 