# Properties

 Label 432450fz Number of curves $6$ Conductor $432450$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("432450.fz1")

sage: E.isogeny_class()

## Elliptic curves in class 432450fz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
432450.fz6 432450fz1 [1, -1, 1, 12968995, -101854831003] [2] 94371840 $$\Gamma_0(N)$$-optimal*
432450.fz5 432450fz2 [1, -1, 1, -263799005, -1558208047003] [2, 2] 188743680 $$\Gamma_0(N)$$-optimal*
432450.fz4 432450fz3 [1, -1, 1, -800037005, 6791017612997] [2] 377487360 $$\Gamma_0(N)$$-optimal*
432450.fz2 432450fz4 [1, -1, 1, -4155849005, -103117360747003] [2, 2] 377487360
432450.fz3 432450fz5 [1, -1, 1, -4090981505, -106492157302003] [2] 754974720
432450.fz1 432450fz6 [1, -1, 1, -66493516505, -6599575041592003] [2] 754974720
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 432450fz1.

## Rank

sage: E.rank()

The elliptic curves in class 432450fz have rank $$0$$.

## Modular form 432450.2.a.fz

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} + 6q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.