# Properties

 Label 2205j Number of curves $8$ Conductor $2205$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 2205j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.i7 2205j1 $$[1, -1, 0, -9, 1728]$$ $$-1/15$$ $$-1286491815$$ $$[2]$$ $$768$$ $$0.42684$$ $$\Gamma_0(N)$$-optimal
2205.i6 2205j2 $$[1, -1, 0, -2214, 40095]$$ $$13997521/225$$ $$19297377225$$ $$[2, 2]$$ $$1536$$ $$0.77341$$
2205.i5 2205j3 $$[1, -1, 0, -4419, -51192]$$ $$111284641/50625$$ $$4341909875625$$ $$[2, 2]$$ $$3072$$ $$1.1200$$
2205.i4 2205j4 $$[1, -1, 0, -35289, 2560410]$$ $$56667352321/15$$ $$1286491815$$ $$[2]$$ $$3072$$ $$1.1200$$
2205.i2 2205j5 $$[1, -1, 0, -59544, -5574717]$$ $$272223782641/164025$$ $$14067787997025$$ $$[2, 2]$$ $$6144$$ $$1.4666$$
2205.i8 2205j6 $$[1, -1, 0, 15426, -396495]$$ $$4733169839/3515625$$ $$-301521519140625$$ $$[2]$$ $$6144$$ $$1.4666$$
2205.i1 2205j7 $$[1, -1, 0, -952569, -357605172]$$ $$1114544804970241/405$$ $$34735279005$$ $$[2]$$ $$12288$$ $$1.8131$$
2205.i3 2205j8 $$[1, -1, 0, -48519, -7711362]$$ $$-147281603041/215233605$$ $$-18459751409696205$$ $$[2]$$ $$12288$$ $$1.8131$$

## Rank

sage: E.rank()

The elliptic curves in class 2205j have rank $$0$$.

## Complex multiplication

The elliptic curves in class 2205j do not have complex multiplication.

## Modular form2205.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} - 3q^{8} + q^{10} + 4q^{11} + 2q^{13} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.