# Properties

 Label 1225.b Number of curves $2$ Conductor $1225$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 1225.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1225.b1 1225h2 $$[1, 1, 1, -208083, -36621194]$$ $$-162677523113838677$$ $$-6125$$ $$[]$$ $$1776$$ $$1.2705$$
1225.b2 1225h1 $$[1, 1, 1, -8, 6]$$ $$-9317$$ $$-6125$$ $$[]$$ $$48$$ $$-0.53497$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1225.b have rank $$1$$.

## Complex multiplication

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

## Modular form1225.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 37 \\ 37 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 