# Properties

 Label 825.b Number of curves $2$ Conductor $825$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 825.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
825.b1 825a1 $$[0, -1, 1, -23, 53]$$ $$-56197120/3267$$ $$-81675$$ $$[]$$ $$72$$ $$-0.30121$$ $$\Gamma_0(N)$$-optimal
825.b2 825a2 $$[0, -1, 1, 127, 38]$$ $$8990228480/5314683$$ $$-132867075$$ $$[]$$ $$216$$ $$0.24810$$

## Rank

sage: E.rank()

The elliptic curves in class 825.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 825.b do not have complex multiplication.

## Modular form825.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{4} + q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} - 7 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 