# Properties

 Label 12480.cm Number of curves $4$ Conductor $12480$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cm1")

sage: E.isogeny_class()

## Elliptic curves in class 12480.cm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12480.cm1 12480x4 $$[0, 1, 0, -33281, 2325855]$$ $$31103978031362/195$$ $$25559040$$ $$$$ $$24576$$ $$1.0270$$
12480.cm2 12480x3 $$[0, 1, 0, -2881, 4895]$$ $$20183398562/11567205$$ $$1516136693760$$ $$$$ $$24576$$ $$1.0270$$
12480.cm3 12480x2 $$[0, 1, 0, -2081, 35775]$$ $$15214885924/38025$$ $$2492006400$$ $$[2, 2]$$ $$12288$$ $$0.68038$$
12480.cm4 12480x1 $$[0, 1, 0, -81, 975]$$ $$-3631696/24375$$ $$-399360000$$ $$$$ $$6144$$ $$0.33381$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 12480.cm have rank $$0$$.

## Complex multiplication

The elliptic curves in class 12480.cm do not have complex multiplication.

## Modular form 12480.2.a.cm

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + 4q^{7} + q^{9} - 4q^{11} - q^{13} - q^{15} + 6q^{17} + O(q^{20})$$ ## 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. 