# Properties

 Label 275.a Number of curves $4$ Conductor $275$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 275.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
275.a1 275a4 $$[1, -1, 1, -1480, 22272]$$ $$22930509321/6875$$ $$107421875$$ $$$$ $$96$$ $$0.51905$$
275.a2 275a3 $$[1, -1, 1, -730, -7228]$$ $$2749884201/73205$$ $$1143828125$$ $$$$ $$96$$ $$0.51905$$
275.a3 275a2 $$[1, -1, 1, -105, 272]$$ $$8120601/3025$$ $$47265625$$ $$[2, 2]$$ $$48$$ $$0.17248$$
275.a4 275a1 $$[1, -1, 1, 20, 22]$$ $$59319/55$$ $$-859375$$ $$$$ $$24$$ $$-0.17409$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 275.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 275.a do not have complex multiplication.

## Modular form275.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} - q^{11} - 2 q^{13} - q^{16} - 6 q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 