# Properties

 Label 68544ca Number of curves $6$ Conductor $68544$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("68544.bq1")

sage: E.isogeny_class()

## Elliptic curves in class 68544ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68544.bq5 68544ca1 [0, 0, 0, -40460556, -98935513616] [2] 5898240 $$\Gamma_0(N)$$-optimal
68544.bq4 68544ca2 [0, 0, 0, -52257036, -36541571600] [2, 2] 11796480
68544.bq6 68544ca3 [0, 0, 0, 201551604, -287406031376] [2] 23592960
68544.bq2 68544ca4 [0, 0, 0, -494809356, 4207535177200] [2, 2] 23592960
68544.bq3 68544ca5 [0, 0, 0, -168539916, 9673331343856] [2] 47185920
68544.bq1 68544ca6 [0, 0, 0, -7901915916, 270362650933744] [2] 47185920

## Rank

sage: E.rank()

The elliptic curves in class 68544ca have rank $$1$$.

## Modular form 68544.2.a.bq

sage: E.q_eigenform(10)

$$q - 2q^{5} + q^{7} + 4q^{11} + 2q^{13} - q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.