# Properties

 Label 14400x Number of curves $6$ Conductor $14400$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 14400x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14400.cy5 14400x1 $$[0, 0, 0, 600, -7000]$$ $$2048/3$$ $$-34992000000$$ $$$$ $$8192$$ $$0.70867$$ $$\Gamma_0(N)$$-optimal
14400.cy4 14400x2 $$[0, 0, 0, -3900, -70000]$$ $$35152/9$$ $$1679616000000$$ $$[2, 2]$$ $$16384$$ $$1.0552$$
14400.cy2 14400x3 $$[0, 0, 0, -57900, -5362000]$$ $$28756228/3$$ $$2239488000000$$ $$$$ $$32768$$ $$1.4018$$
14400.cy3 14400x4 $$[0, 0, 0, -21900, 1190000]$$ $$1556068/81$$ $$60466176000000$$ $$[2, 2]$$ $$32768$$ $$1.4018$$
14400.cy1 14400x5 $$[0, 0, 0, -345900, 78302000]$$ $$3065617154/9$$ $$13436928000000$$ $$$$ $$65536$$ $$1.7484$$
14400.cy6 14400x6 $$[0, 0, 0, 14100, 4718000]$$ $$207646/6561$$ $$-9795520512000000$$ $$$$ $$65536$$ $$1.7484$$

## Rank

sage: E.rank()

The elliptic curves in class 14400x have rank $$0$$.

## Complex multiplication

The elliptic curves in class 14400x do not have complex multiplication.

## Modular form 14400.2.a.x

sage: E.q_eigenform(10)

$$q + 4q^{11} - 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 