# Properties

 Label 414.d Number of curves $2$ Conductor $414$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 414.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414.d1 414b2 $$[1, -1, 1, -284, -1767]$$ $$3463512697/3174$$ $$2313846$$ $$$$ $$128$$ $$0.14475$$
414.d2 414b1 $$[1, -1, 1, -14, -39]$$ $$-389017/828$$ $$-603612$$ $$$$ $$64$$ $$-0.20182$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 414.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 414.d do not have complex multiplication.

## Modular form414.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8} + 2 q^{10} + 6 q^{11} - 2 q^{13} - 2 q^{14} + q^{16} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 