# Properties

 Label 388416fi Number of curves $6$ Conductor $388416$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("388416.fi1")

sage: E.isogeny_class()

## Elliptic curves in class 388416fi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388416.fi5 388416fi1 [0, 1, 0, -1299233409, -18003032277633] [2] 212336640 $$\Gamma_0(N)$$-optimal
388416.fi4 388416fi2 [0, 1, 0, -1678031489, -6649771983489] [2, 2] 424673280
388416.fi2 388416fi3 [0, 1, 0, -15888878209, 765610271321471] [2, 2] 849346560
388416.fi6 388416fi4 [0, 1, 0, 6472045951, -52295095693953] [2] 849346560
388416.fi1 388416fi5 [0, 1, 0, -253739299969, 49195904458658687] [2] 1698693120
388416.fi3 388416fi6 [0, 1, 0, -5412003969, 1760186229049215] [2] 1698693120

## Rank

sage: E.rank()

The elliptic curves in class 388416fi have rank $$0$$.

## Modular form 388416.2.a.fi

sage: E.q_eigenform(10)

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

## 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.