# Properties

 Label 344850fs Number of curves $4$ Conductor $344850$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("344850.fs1")

sage: E.isogeny_class()

## Elliptic curves in class 344850fs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
344850.fs4 344850fs1 [1, 1, 1, -30313, -3478969]  2949120 $$\Gamma_0(N)$$-optimal
344850.fs3 344850fs2 [1, 1, 1, -574813, -167917969] [2, 2] 5898240
344850.fs2 344850fs3 [1, 1, 1, -665563, -111471469]  11796480
344850.fs1 344850fs4 [1, 1, 1, -9196063, -10737570469]  11796480

## Rank

sage: E.rank()

The elliptic curves in class 344850fs have rank $$0$$.

## Modular form 344850.2.a.fs

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + 4q^{7} + q^{8} + q^{9} - q^{12} - 2q^{13} + 4q^{14} + q^{16} - 2q^{17} + q^{18} + q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 