# Properties

 Label 463680ko Number of curves $6$ Conductor $463680$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ko1")

sage: E.isogeny_class()

## Elliptic curves in class 463680ko

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.ko6 463680ko1 $$[0, 0, 0, 82968, 2860216]$$ $$84611246065664/53699121315$$ $$-40086179265162240$$ $$[2]$$ $$3670016$$ $$1.8748$$ $$\Gamma_0(N)$$-optimal*
463680.ko5 463680ko2 $$[0, 0, 0, -349212, 23431984]$$ $$394315384276816/208332909225$$ $$2488314934477209600$$ $$[2, 2]$$ $$7340032$$ $$2.2214$$ $$\Gamma_0(N)$$-optimal*
463680.ko2 463680ko3 $$[0, 0, 0, -4406412, 3556441744]$$ $$198048499826486404/242568272835$$ $$11588879705487114240$$ $$[2]$$ $$14680064$$ $$2.5680$$ $$\Gamma_0(N)$$-optimal*
463680.ko3 463680ko4 $$[0, 0, 0, -3206892, -2192984624]$$ $$76343005935514084/694180580625$$ $$33164993709711360000$$ $$[2, 2]$$ $$14680064$$ $$2.5680$$
463680.ko4 463680ko5 $$[0, 0, 0, -938892, -5234826224]$$ $$-957928673903042/123339801817575$$ $$-11785301593294395801600$$ $$[2]$$ $$29360128$$ $$2.9145$$
463680.ko1 463680ko6 $$[0, 0, 0, -51197772, -141001805936]$$ $$155324313723954725282/13018359375$$ $$1243923609600000000$$ $$[2]$$ $$29360128$$ $$2.9145$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 463680ko1.

## Rank

sage: E.rank()

The elliptic curves in class 463680ko have rank $$0$$.

## Complex multiplication

The elliptic curves in class 463680ko do not have complex multiplication.

## Modular form 463680.2.a.ko

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} + 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.