# Properties

 Label 226512.fd Number of curves $6$ Conductor $226512$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("226512.fd1")

sage: E.isogeny_class()

## Elliptic curves in class 226512.fd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
226512.fd1 226512cq4 [0, 0, 0, -119598699, -503428498342] [2] 15728640
226512.fd2 226512cq5 [0, 0, 0, -52429179, 141492303722] [2] 31457280
226512.fd3 226512cq3 [0, 0, 0, -8259339, -6114467590] [2, 2] 15728640
226512.fd4 226512cq2 [0, 0, 0, -7475259, -7865318230] [2, 2] 7864320
226512.fd5 226512cq1 [0, 0, 0, -418539, -149500582] [2] 3932160 $$\Gamma_0(N)$$-optimal
226512.fd6 226512cq6 [0, 0, 0, 23365221, -41666797942] [2] 31457280

## Rank

sage: E.rank()

The elliptic curves in class 226512.fd have rank $$0$$.

## Modular form 226512.2.a.fd

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.