# Properties

 Label 135240.e Number of curves $6$ Conductor $135240$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 135240.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.e1 135240bs5 $$[0, -1, 0, -69685856, 223928874156]$$ $$155324313723954725282/13018359375$$ $$3136710578400000000$$ $$[2]$$ $$11010048$$ $$2.9916$$
135240.e2 135240bs4 $$[0, -1, 0, -5997616, -5645498564]$$ $$198048499826486404/242568272835$$ $$29222824684303272960$$ $$[2]$$ $$5505024$$ $$2.6450$$
135240.e3 135240bs3 $$[0, -1, 0, -4364936, 3483833340]$$ $$76343005935514084/694180580625$$ $$83629722757069440000$$ $$[2, 2]$$ $$5505024$$ $$2.6450$$
135240.e4 135240bs6 $$[0, -1, 0, -1277936, 8313136140]$$ $$-957928673903042/123339801817575$$ $$-29718127296585484646400$$ $$[2]$$ $$11010048$$ $$2.9916$$
135240.e5 135240bs2 $$[0, -1, 0, -475316, -37050684]$$ $$394315384276816/208332909225$$ $$6274600559977478400$$ $$[2, 2]$$ $$2752512$$ $$2.2985$$
135240.e6 135240bs1 $$[0, -1, 0, 112929, -4579560]$$ $$84611246065664/53699121315$$ $$-101082366777414960$$ $$[2]$$ $$1376256$$ $$1.9519$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 135240.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 135240.e do not have complex multiplication.

## Modular form 135240.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.