# Properties

 Label 370.d Number of curves $4$ Conductor $370$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 370.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
370.d1 370d3 $$[1, 0, 0, -5275, -147903]$$ $$16232905099479601/4052240$$ $$4052240$$ $$$$ $$288$$ $$0.64351$$
370.d2 370d4 $$[1, 0, 0, -5255, -149075]$$ $$-16048965315233521/256572640900$$ $$-256572640900$$ $$$$ $$576$$ $$0.99008$$
370.d3 370d1 $$[1, 0, 0, -75, -143]$$ $$46694890801/18944000$$ $$18944000$$ $$$$ $$96$$ $$0.094202$$ $$\Gamma_0(N)$$-optimal
370.d4 370d2 $$[1, 0, 0, 245, -975]$$ $$1625964918479/1369000000$$ $$-1369000000$$ $$$$ $$192$$ $$0.44078$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form370.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 