# Properties

 Label 4140.a Number of curves $4$ Conductor $4140$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 4140.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4140.a1 4140e3 $$[0, 0, 0, -109488, 13944337]$$ $$12444451776495616/912525$$ $$10643691600$$ $$[6]$$ $$12096$$ $$1.3748$$
4140.a2 4140e4 $$[0, 0, 0, -109263, 14004502]$$ $$-772993034343376/6661615005$$ $$-1243217238693120$$ $$[6]$$ $$24192$$ $$1.7213$$
4140.a3 4140e1 $$[0, 0, 0, -1488, 15037]$$ $$31238127616/9703125$$ $$113177250000$$ $$[2]$$ $$4032$$ $$0.82545$$ $$\Gamma_0(N)$$-optimal
4140.a4 4140e2 $$[0, 0, 0, 4137, 101662]$$ $$41957807024/48205125$$ $$-8996233248000$$ $$[2]$$ $$8064$$ $$1.1720$$

## Rank

sage: E.rank()

The elliptic curves in class 4140.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 4140.a do not have complex multiplication.

## Modular form4140.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.