# Properties

 Label 120.a Number of curves $4$ Conductor $120$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 120.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
120.a1 120b3 $$[0, 1, 0, -216, -1296]$$ $$546718898/405$$ $$829440$$ $$$$ $$32$$ $$0.069403$$
120.a2 120b4 $$[0, 1, 0, -136, 560]$$ $$136835858/1875$$ $$3840000$$ $$$$ $$32$$ $$0.069403$$
120.a3 120b2 $$[0, 1, 0, -16, -16]$$ $$470596/225$$ $$230400$$ $$[2, 2]$$ $$16$$ $$-0.27717$$
120.a4 120b1 $$[0, 1, 0, 4, 0]$$ $$21296/15$$ $$-3840$$ $$$$ $$8$$ $$-0.62374$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 120.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 120.a do not have complex multiplication.

## Modular form120.2.a.a

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + 4 q^{7} + q^{9} - 6 q^{13} - q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 