# Properties

 Label 259920c Number of curves $4$ Conductor $259920$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("259920.c1")

sage: E.isogeny_class()

## Elliptic curves in class 259920c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
259920.c4 259920c1 [0, 0, 0, -520923, -246937718]  6635520 $$\Gamma_0(N)$$-optimal
259920.c3 259920c2 [0, 0, 0, -9878043, -11945209142] [2, 2] 13271040
259920.c2 259920c3 [0, 0, 0, -11437563, -7921335638]  26542080
259920.c1 259920c4 [0, 0, 0, -158032443, -764658453782]  26542080

## Rank

sage: E.rank()

The elliptic curves in class 259920c have rank $$0$$.

## Modular form 259920.2.a.c

sage: E.q_eigenform(10)

$$q - q^{5} - 4q^{7} - 4q^{11} + 2q^{13} + 2q^{17} + O(q^{20})$$

## 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. 