# Properties

 Label 1225a Number of curves $3$ Conductor $1225$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 1225a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1225.e2 1225a1 $$[0, 1, 1, -1633, -28731]$$ $$-262144/35$$ $$-64339296875$$ $$[]$$ $$768$$ $$0.80652$$ $$\Gamma_0(N)$$-optimal
1225.e3 1225a2 $$[0, 1, 1, 10617, 75394]$$ $$71991296/42875$$ $$-78815638671875$$ $$[]$$ $$2304$$ $$1.3558$$
1225.e1 1225a3 $$[0, 1, 1, -160883, 25929019]$$ $$-250523582464/13671875$$ $$-25132537841796875$$ $$[]$$ $$6912$$ $$1.9051$$

## Rank

sage: E.rank()

The elliptic curves in class 1225a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1225a do not have complex multiplication.

## Modular form1225.2.a.a

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} + 4q^{16} + 3q^{17} - 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 