# Properties

 Label 344850.gz Number of curves $4$ Conductor $344850$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("344850.gz1")

sage: E.isogeny_class()

## Elliptic curves in class 344850.gz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
344850.gz1 344850gz3 [1, 0, 0, -1878588, 989452542]  7864320
344850.gz2 344850gz2 [1, 0, 0, -154338, 4905792] [2, 2] 3932160
344850.gz3 344850gz1 [1, 0, 0, -93838, -11005708]  1966080 $$\Gamma_0(N)$$-optimal
344850.gz4 344850gz4 [1, 0, 0, 601912, 38937042]  7864320

## Rank

sage: E.rank()

The elliptic curves in class 344850.gz have rank $$1$$.

## Modular form 344850.2.a.gz

sage: E.q_eigenform(10)

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

## 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 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. 