# Properties

 Label 38640.f Number of curves $4$ Conductor $38640$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.f1 38640bj4 $$[0, -1, 0, -45536, -3195264]$$ $$2549399737314529/388286718750$$ $$1590422400000000$$ $$$$ $$196608$$ $$1.6407$$
38640.f2 38640bj2 $$[0, -1, 0, -12416, 487680]$$ $$51682540549249/5249002500$$ $$21499914240000$$ $$[2, 2]$$ $$98304$$ $$1.2941$$
38640.f3 38640bj1 $$[0, -1, 0, -12096, 516096]$$ $$47788676405569/579600$$ $$2374041600$$ $$$$ $$49152$$ $$0.94755$$ $$\Gamma_0(N)$$-optimal
38640.f4 38640bj3 $$[0, -1, 0, 15584, 2346880]$$ $$102181603702751/642612880350$$ $$-2632142357913600$$ $$$$ $$196608$$ $$1.6407$$

## Rank

sage: E.rank()

The elliptic curves in class 38640.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 38640.f do not have complex multiplication.

## Modular form 38640.2.a.f

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - q^{7} + q^{9} - 6q^{13} + q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 