# Properties

 Label 370.a Number of curves $3$ Conductor $370$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 370.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
370.a1 370c3 $$[1, 0, 1, -54, 146]$$ $$-16954786009/370$$ $$-370$$ $$$$ $$324$$ $$-0.39121$$
370.a2 370c1 $$[1, 0, 1, -19, 342]$$ $$-702595369/50653000$$ $$-50653000$$ $$$$ $$108$$ $$0.15809$$ $$\Gamma_0(N)$$-optimal
370.a3 370c2 $$[1, 0, 1, 166, -9204]$$ $$510273943271/37000000000$$ $$-37000000000$$ $$[]$$ $$324$$ $$0.70740$$

## Rank

sage: E.rank()

The elliptic curves in class 370.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 370.a do not have complex multiplication.

## Modular form370.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + 3 q^{11} - 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + q^{16} + 3 q^{17} - q^{18} + 2 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 