# Properties

 Label 444360p Number of curves $6$ Conductor $444360$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("444360.p1")

sage: E.isogeny_class()

## Elliptic curves in class 444360p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
444360.p4 444360p1 [0, -1, 0, -92751, -10841424] [2] 1622016 $$\Gamma_0(N)$$-optimal*
444360.p3 444360p2 [0, -1, 0, -95396, -10187580] [2, 2] 3244032 $$\Gamma_0(N)$$-optimal*
444360.p2 444360p3 [0, -1, 0, -359896, 72230620] [2, 2] 6488064 $$\Gamma_0(N)$$-optimal*
444360.p5 444360p4 [0, -1, 0, 126784, -50802084] [2] 6488064
444360.p1 444360p5 [0, -1, 0, -5544096, 5026252140] [2] 12976128 $$\Gamma_0(N)$$-optimal*
444360.p6 444360p6 [0, -1, 0, 592304, 388741900] [2] 12976128
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 444360p1.

## Rank

sage: E.rank()

The elliptic curves in class 444360p have rank $$2$$.

## Modular form 444360.2.a.p

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{7} + q^{9} + 4q^{11} - 2q^{13} + q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.