# Properties

 Label 2646.f Number of curves $3$ Conductor $2646$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 2646.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2646.f1 2646l2 $$[1, -1, 0, -52047, -4557281]$$ $$-545407363875/14$$ $$-400241898$$ $$[]$$ $$5184$$ $$1.1668$$
2646.f2 2646l1 $$[1, -1, 0, -597, -7043]$$ $$-7414875/2744$$ $$-8716379112$$ $$[]$$ $$1728$$ $$0.61753$$ $$\Gamma_0(N)$$-optimal
2646.f3 2646l3 $$[1, -1, 0, 4548, 71504]$$ $$4492125/3584$$ $$-8299415996928$$ $$[]$$ $$5184$$ $$1.1668$$

## Rank

sage: E.rank()

The elliptic curves in class 2646.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2646.f do not have complex multiplication.

## Modular form2646.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{8} - 5 q^{13} + q^{16} + 3 q^{17} - 2 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 