# Properties

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

# Related objects

Show commands: SageMath
E = EllipticCurve("p1")

E.isogeny_class()

## Elliptic curves in class 3360.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3360.p1 3360k3 $$[0, 1, 0, -317521, 68760575]$$ $$864335783029582144/59535$$ $$243855360$$ $$[2]$$ $$15360$$ $$1.5103$$
3360.p2 3360k2 $$[0, 1, 0, -22296, 786060]$$ $$2394165105226952/854262178245$$ $$437382235261440$$ $$[2]$$ $$15360$$ $$1.5103$$
3360.p3 3360k1 $$[0, 1, 0, -19846, 1069280]$$ $$13507798771700416/3544416225$$ $$226842638400$$ $$[2, 2]$$ $$7680$$ $$1.1637$$ $$\Gamma_0(N)$$-optimal
3360.p4 3360k4 $$[0, 1, 0, -17416, 1343384]$$ $$-1141100604753992/875529151875$$ $$-448270925760000$$ $$[2]$$ $$15360$$ $$1.5103$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3360.2.a.p

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + q^{7} + q^{9} - 4 q^{11} - 6 q^{13} - q^{15} + 6 q^{17} + O(q^{20})$$

## 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.