# Properties

 Label 3360.d Number of curves $4$ Conductor $3360$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 3360.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3360.d1 3360b3 $$[0, -1, 0, -324016, 71098216]$$ $$7347751505995469192/72930375$$ $$37340352000$$ $$[2]$$ $$15360$$ $$1.6049$$
3360.d2 3360b2 $$[0, -1, 0, -29016, 67716]$$ $$5276930158229192/3050936350875$$ $$1562079411648000$$ $$[2]$$ $$15360$$ $$1.6049$$
3360.d3 3360b1 $$[0, -1, 0, -20266, 1114216]$$ $$14383655824793536/45209390625$$ $$2893401000000$$ $$[2, 2]$$ $$7680$$ $$1.2583$$ $$\Gamma_0(N)$$-optimal
3360.d4 3360b4 $$[0, -1, 0, -11761, 2048065]$$ $$-43927191786304/415283203125$$ $$-1701000000000000$$ $$[2]$$ $$15360$$ $$1.6049$$

## Rank

sage: E.rank()

The elliptic curves in class 3360.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3360.d do not have complex multiplication.

## Modular form3360.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - q^{7} + q^{9} + 4q^{11} - 2q^{13} + q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.