Properties

 Label 443760df Number of curves $4$ Conductor $443760$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

Elliptic curves in class 443760df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
443760.df3 443760df1 [0, 1, 0, -2027120, -1110163500] [2] 8515584 $$\Gamma_0(N)$$-optimal*
443760.df2 443760df2 [0, 1, 0, -2618800, -409851052] [2, 2] 17031168 $$\Gamma_0(N)$$-optimal*
443760.df1 443760df3 [0, 1, 0, -24806800, 47223347348] [2] 34062336 $$\Gamma_0(N)$$-optimal*
443760.df4 443760df4 [0, 1, 0, 10102320, -3213585900] [4] 34062336
*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 443760df1.

Rank

sage: E.rank()

The elliptic curves in class 443760df have rank $$1$$.

Complex multiplication

The elliptic curves in class 443760df do not have complex multiplication.

Modular form 443760.2.a.df

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + 4q^{7} + q^{9} - 2q^{13} + q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.