# Properties

 Label 86394.d Number of curves $6$ Conductor $86394$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("86394.d1")

sage: E.isogeny_class()

## Elliptic curves in class 86394.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86394.d1 86394b6 [1, 1, 0, -1659951086, 26030318798796]  39321600
86394.d2 86394b4 [1, 1, 0, -103944326, 405065871060] [2, 2] 19660800
86394.d3 86394b5 [1, 1, 0, -35405086, 931351279324]  39321600
86394.d4 86394b2 [1, 1, 0, -10977606, -3522863340] [2, 2] 9830400
86394.d5 86394b1 [1, 1, 0, -8499526, -9529233644]  4915200 $$\Gamma_0(N)$$-optimal
86394.d6 86394b3 [1, 1, 0, 42339834, -27654336684]  19660800

## Rank

sage: E.rank()

The elliptic curves in class 86394.d have rank $$0$$.

## Modular form 86394.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 2q^{10} - q^{12} + 2q^{13} + q^{14} + 2q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 