# Properties

 Label 14400ef Number of curves $8$ Conductor $14400$ CM no Rank $1$ Graph

# Learn more

Show commands for: SageMath
sage: E = EllipticCurve("ef1")

sage: E.isogeny_class()

## Elliptic curves in class 14400ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14400.o8 14400ef1 $$[0, 0, 0, 21300, 3674000]$$ $$357911/2160$$ $$-6449725440000000$$ $$[2]$$ $$73728$$ $$1.7155$$ $$\Gamma_0(N)$$-optimal
14400.o6 14400ef2 $$[0, 0, 0, -266700, 48026000]$$ $$702595369/72900$$ $$217678233600000000$$ $$[2, 2]$$ $$147456$$ $$2.0620$$
14400.o7 14400ef3 $$[0, 0, 0, -194700, -108214000]$$ $$-273359449/1536000$$ $$-4586471424000000000$$ $$[2]$$ $$221184$$ $$2.2648$$
14400.o5 14400ef4 $$[0, 0, 0, -986700, -324934000]$$ $$35578826569/5314410$$ $$15868743229440000000$$ $$[2]$$ $$294912$$ $$2.4086$$
14400.o4 14400ef5 $$[0, 0, 0, -4154700, 3259514000]$$ $$2656166199049/33750$$ $$100776960000000000$$ $$[2]$$ $$294912$$ $$2.4086$$
14400.o3 14400ef6 $$[0, 0, 0, -4802700, -4043446000]$$ $$4102915888729/9000000$$ $$26873856000000000000$$ $$[2, 2]$$ $$442368$$ $$2.6113$$
14400.o1 14400ef7 $$[0, 0, 0, -76802700, -259067446000]$$ $$16778985534208729/81000$$ $$241864704000000000$$ $$[2]$$ $$884736$$ $$2.9579$$
14400.o2 14400ef8 $$[0, 0, 0, -6530700, -874294000]$$ $$10316097499609/5859375000$$ $$17496000000000000000000$$ $$[2]$$ $$884736$$ $$2.9579$$

## Rank

sage: E.rank()

The elliptic curves in class 14400ef have rank $$1$$.

## Complex multiplication

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

## Modular form 14400.2.a.ef

sage: E.q_eigenform(10)

$$q - 4q^{7} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.