# Properties

 Label 302016df Number of curves $6$ Conductor $302016$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("302016.df1")

sage: E.isogeny_class()

## Elliptic curves in class 302016df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
302016.df5 302016df1 [0, -1, 0, -186017, -44234463] [2] 3932160 $$\Gamma_0(N)$$-optimal
302016.df4 302016df2 [0, -1, 0, -3322337, -2329357215] [2, 2] 7864320
302016.df3 302016df3 [0, -1, 0, -3670817, -1810470495] [2, 2] 15728640
302016.df1 302016df4 [0, -1, 0, -53154977, -149146281183] [2] 15728640
302016.df2 302016df5 [0, -1, 0, -23301857, 41931412833] [2] 31457280
302016.df6 302016df6 [0, -1, 0, 10384543, -12349179423] [2] 31457280

## Rank

sage: E.rank()

The elliptic curves in class 302016df have rank $$1$$.

## Modular form 302016.2.a.df

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} + q^{13} - 2q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.