# Properties

 Label 5225.b Number of curves $2$ Conductor $5225$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 5225.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5225.b1 5225c1 $$[0, -1, 1, -683, 8268]$$ $$-2258403328/480491$$ $$-7507671875$$ $$[]$$ $$2592$$ $$0.61630$$ $$\Gamma_0(N)$$-optimal
5225.b2 5225c2 $$[0, -1, 1, 4817, -48107]$$ $$790939860992/517504691$$ $$-8086010796875$$ $$[]$$ $$7776$$ $$1.1656$$

## Rank

sage: E.rank()

The elliptic curves in class 5225.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5225.b do not have complex multiplication.

## Modular form5225.2.a.b

sage: E.q_eigenform(10)

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

## 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. 