# Properties

 Label 225e Number of curves $2$ Conductor $225$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 225e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
225.a1 225e1 $$[0, 0, 1, -75, 256]$$ $$-102400/3$$ $$-1366875$$ $$[]$$ $$48$$ $$-0.043792$$ $$\Gamma_0(N)$$-optimal
225.a2 225e2 $$[0, 0, 1, 375, -12344]$$ $$20480/243$$ $$-69198046875$$ $$[]$$ $$240$$ $$0.76093$$

## Rank

sage: E.rank()

The elliptic curves in class 225e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 225e do not have complex multiplication.

## Modular form225.2.a.e

sage: E.q_eigenform(10)

$$q - 2 q^{2} + 2 q^{4} - 3 q^{7} - 2 q^{11} + q^{13} + 6 q^{14} - 4 q^{16} - 2 q^{17} - 5 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.