# Properties

 Label 129360.fj Number of curves $6$ Conductor $129360$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("fj1")

sage: E.isogeny_class()

## Elliptic curves in class 129360.fj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.fj1 129360gk6 $$[0, 1, 0, -2390572816, -44989251846316]$$ $$3135316978843283198764801/571725$$ $$275508734054400$$ $$[2]$$ $$23592960$$ $$3.5652$$
129360.fj2 129360gk4 $$[0, 1, 0, -149410816, -702994261516]$$ $$765458482133960722801/326869475625$$ $$157515230977251840000$$ $$[2, 2]$$ $$11796480$$ $$3.2187$$
129360.fj3 129360gk5 $$[0, 1, 0, -148669936, -710310303340]$$ $$-754127868744065783521/15825714261328125$$ $$-7626258256408545600000000$$ $$[2]$$ $$23592960$$ $$3.5652$$
129360.fj4 129360gk3 $$[0, 1, 0, -19948896, 18078514740]$$ $$1821931919215868881/761147600816295$$ $$366789648746235045703680$$ $$[2]$$ $$11796480$$ $$3.2187$$
129360.fj5 129360gk2 $$[0, 1, 0, -9384496, -10872167020]$$ $$189674274234120481/3859869269025$$ $$1860033575450715033600$$ $$[2, 2]$$ $$5898240$$ $$2.8721$$
129360.fj6 129360gk1 $$[0, 1, 0, 27424, -507760716]$$ $$4733169839/231139696095$$ $$-111383978417687162880$$ $$[2]$$ $$2949120$$ $$2.5255$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 129360.fj have rank $$1$$.

## Complex multiplication

The elliptic curves in class 129360.fj do not have complex multiplication.

## Modular form 129360.2.a.fj

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.