# Properties

 Label 455175ce Number of curves 4 Conductor 455175 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("455175.ce1")

sage: E.isogeny_class()

## Elliptic curves in class 455175ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
455175.ce4 455175ce1 [1, -1, 1, 908995, -561249628] [2] 14155776 $$\Gamma_0(N)$$-optimal*
455175.ce3 455175ce2 [1, -1, 1, -7219130, -6120887128] [2, 2] 28311552 $$\Gamma_0(N)$$-optimal*
455175.ce2 455175ce3 [1, -1, 1, -34854755, 73635526622] [2] 56623104 $$\Gamma_0(N)$$-optimal*
455175.ce1 455175ce4 [1, -1, 1, -109633505, -441791638378] [2] 56623104
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 455175ce1.

## Rank

sage: E.rank()

The elliptic curves in class 455175ce have rank $$1$$.

## Modular form 455175.2.a.ce

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + q^{7} + 3q^{8} + 6q^{13} - q^{14} - q^{16} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.