Properties

 Label 289800cf Number of curves $6$ Conductor $289800$ CM no Rank $1$ Graph

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Show commands: SageMath
sage: E = EllipticCurve("cf1")

sage: E.isogeny_class()

Elliptic curves in class 289800cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
289800.cf6 289800cf1 $$[0, 0, 0, 518550, -44690875]$$ $$84611246065664/53699121315$$ $$-9786664859658750000$$ $$[2]$$ $$5505024$$ $$2.3330$$ $$\Gamma_0(N)$$-optimal
289800.cf5 289800cf2 $$[0, 0, 0, -2182575, -366124750]$$ $$394315384276816/208332909225$$ $$607498763300100000000$$ $$[2, 2]$$ $$11010048$$ $$2.6795$$
289800.cf3 289800cf3 $$[0, 0, 0, -20043075, 34265384750]$$ $$76343005935514084/694180580625$$ $$8096922292410000000000$$ $$[2, 2]$$ $$22020096$$ $$3.0261$$
289800.cf2 289800cf4 $$[0, 0, 0, -27540075, -55569402250]$$ $$198048499826486404/242568272835$$ $$2829316334347440000000$$ $$[2]$$ $$22020096$$ $$3.0261$$
289800.cf1 289800cf5 $$[0, 0, 0, -319986075, 2203153217750]$$ $$155324313723954725282/13018359375$$ $$303692287500000000000$$ $$[2]$$ $$44040192$$ $$3.3727$$
289800.cf4 289800cf6 $$[0, 0, 0, -5868075, 81794159750]$$ $$-957928673903042/123339801817575$$ $$-2877270896800389600000000$$ $$[2]$$ $$44040192$$ $$3.3727$$

Rank

sage: E.rank()

The elliptic curves in class 289800cf have rank $$1$$.

Complex multiplication

The elliptic curves in class 289800cf do not have complex multiplication.

Modular form 289800.2.a.cf

sage: E.q_eigenform(10)

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.