# Properties

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

Show commands: SageMath
E = EllipticCurve("c1")

E.isogeny_class()

## Elliptic curves in class 5225.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5225.c1 5225a2 $$[1, 1, 0, -775, 7500]$$ $$3301293169/218405$$ $$3412578125$$ $$$$ $$3072$$ $$0.57868$$
5225.c2 5225a1 $$[1, 1, 0, -150, -625]$$ $$24137569/5225$$ $$81640625$$ $$$$ $$1536$$ $$0.23210$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form5225.2.a.c

sage: E.q_eigenform(10)

$$q + q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + q^{9} - q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{14} - q^{16} + q^{18} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 