# Properties

 Label 58835.f Number of curves $3$ Conductor $58835$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 58835.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58835.f1 58835a3 $$[0, -1, 1, -220771, -41693174]$$ $$-250523582464/13671875$$ $$-64942831419921875$$ $$[]$$ $$388800$$ $$1.9842$$
58835.f2 58835a1 $$[0, -1, 1, -2241, 46056]$$ $$-262144/35$$ $$-166253648435$$ $$[]$$ $$43200$$ $$0.88564$$ $$\Gamma_0(N)$$-optimal
58835.f3 58835a2 $$[0, -1, 1, 14569, -120363]$$ $$71991296/42875$$ $$-203660719332875$$ $$[]$$ $$129600$$ $$1.4349$$

## Rank

sage: E.rank()

The elliptic curves in class 58835.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 58835.f do not have complex multiplication.

## Modular form 58835.2.a.f

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 