# Properties

 Label 49.a Number of curves $4$ Conductor $49$ CM $$\Q(\sqrt{-7})$$ Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 49.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
49.a1 49a4 $$[1, -1, 0, -1822, 30393]$$ $$16581375$$ $$40353607$$ $$$$ $$14$$ $$0.52039$$   $$-28$$
49.a2 49a3 $$[1, -1, 0, -107, 552]$$ $$-3375$$ $$-40353607$$ $$$$ $$7$$ $$0.17382$$   $$-7$$
49.a3 49a2 $$[1, -1, 0, -37, -78]$$ $$16581375$$ $$343$$ $$$$ $$2$$ $$-0.45256$$   $$-28$$
49.a4 49a1 $$[1, -1, 0, -2, -1]$$ $$-3375$$ $$-343$$ $$$$ $$1$$ $$-0.79914$$ $$\Gamma_0(N)$$-optimal $$-7$$

## Rank

sage: E.rank()

The elliptic curves in class 49.a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 49.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-7})$$.

## Modular form49.2.a.a

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 