# Properties

 Label 215475p Number of curves $6$ Conductor $215475$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("215475.t1")

sage: E.isogeny_class()

## Elliptic curves in class 215475p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
215475.t4 215475p1 [1, 0, 0, -2277363, 1322615592] [2] 2752512 $$\Gamma_0(N)$$-optimal
215475.t3 215475p2 [1, 0, 0, -2298488, 1296821967] [2, 2] 5505024
215475.t5 215475p3 [1, 0, 0, 933637, 4654999842] [2] 11010048
215475.t2 215475p4 [1, 0, 0, -5868613, -3712063408] [2, 2] 11010048
215475.t6 215475p5 [1, 0, 0, 16376012, -25133637283] [2] 22020096
215475.t1 215475p6 [1, 0, 0, -85235238, -302844873033] [2] 22020096

## Rank

sage: E.rank()

The elliptic curves in class 215475p have rank $$0$$.

## Modular form 215475.2.a.t

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} - 4q^{11} - q^{12} - q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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 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.