# Properties

 Label 42.a Number of curves $6$ Conductor $42$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 42.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42.a1 42a4 $$[1, 1, 1, -1344, 18405]$$ $$268498407453697/252$$ $$252$$ $$$$ $$16$$ $$0.18910$$
42.a2 42a5 $$[1, 1, 1, -914, -10915]$$ $$84448510979617/933897762$$ $$933897762$$ $$$$ $$32$$ $$0.53568$$
42.a3 42a3 $$[1, 1, 1, -104, 101]$$ $$124475734657/63011844$$ $$63011844$$ $$[2, 2]$$ $$16$$ $$0.18910$$
42.a4 42a2 $$[1, 1, 1, -84, 261]$$ $$65597103937/63504$$ $$63504$$ $$[2, 4]$$ $$8$$ $$-0.15747$$
42.a5 42a1 $$[1, 1, 1, -4, 5]$$ $$-7189057/16128$$ $$-16128$$ $$$$ $$4$$ $$-0.50405$$ $$\Gamma_0(N)$$-optimal
42.a6 42a6 $$[1, 1, 1, 386, 1277]$$ $$6359387729183/4218578658$$ $$-4218578658$$ $$$$ $$32$$ $$0.53568$$

## Rank

sage: E.rank()

The elliptic curves in class 42.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 42.a do not have complex multiplication.

## Modular form42.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} + 6q^{13} - q^{14} + 2q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 