# Properties

 Label 425880.eb Number of curves $6$ Conductor $425880$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("425880.eb1")

sage: E.isogeny_class()

## Elliptic curves in class 425880.eb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
425880.eb1 425880eb5 [0, 0, 0, -15940587, 24495983174] [2] 18874368 $$\Gamma_0(N)$$-optimal*
425880.eb2 425880eb3 [0, 0, 0, -1034787, 351568334] [2, 2] 9437184 $$\Gamma_0(N)$$-optimal*
425880.eb3 425880eb2 [0, 0, 0, -274287, -49823566] [2, 2] 4718592 $$\Gamma_0(N)$$-optimal*
425880.eb4 425880eb1 [0, 0, 0, -266682, -53007019] [2] 2359296 $$\Gamma_0(N)$$-optimal*
425880.eb5 425880eb4 [0, 0, 0, 364533, -247474474] [2] 9437184
425880.eb6 425880eb6 [0, 0, 0, 1703013, 1896235094] [2] 18874368
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 425880.eb4.

## Rank

sage: E.rank()

The elliptic curves in class 425880.eb have rank $$1$$.

## Modular form 425880.2.a.eb

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.