# Properties

 Label 144.a Number of curves $4$ Conductor $144$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 144.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
144.a1 144a4 $$[0, 0, 0, -135, 594]$$ $$54000$$ $$5038848$$ $$$$ $$24$$ $$0.080464$$   $$-12$$
144.a2 144a2 $$[0, 0, 0, -15, -22]$$ $$54000$$ $$6912$$ $$$$ $$8$$ $$-0.46884$$   $$-12$$
144.a3 144a1 $$[0, 0, 0, 0, -1]$$ $$0$$ $$-432$$ $$$$ $$4$$ $$-0.81542$$ $$\Gamma_0(N)$$-optimal $$-3$$
144.a4 144a3 $$[0, 0, 0, 0, 27]$$ $$0$$ $$-314928$$ $$$$ $$12$$ $$-0.26611$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 144.a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 144.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form144.2.a.a

sage: E.q_eigenform(10)

$$q + 4q^{7} + 2q^{13} - 8q^{19} + O(q^{20})$$ ## 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. 