# Properties

 Label 8400bp Number of curves $6$ Conductor $8400$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("8400.p1")

sage: E.isogeny_class()

## Elliptic curves in class 8400bp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.p6 8400bp1 [0, -1, 0, 3992, 134512] [2] 18432 $$\Gamma_0(N)$$-optimal
8400.p5 8400bp2 [0, -1, 0, -28008, 1414512] [2, 2] 36864
8400.p4 8400bp3 [0, -1, 0, -148008, -20665488] [2] 73728
8400.p2 8400bp4 [0, -1, 0, -420008, 104902512] [2, 2] 73728
8400.p1 8400bp5 [0, -1, 0, -6720008, 6707302512] [2] 147456
8400.p3 8400bp6 [0, -1, 0, -392008, 119462512] [2] 147456

## Rank

sage: E.rank()

The elliptic curves in class 8400bp have rank $$1$$.

## Modular form8400.2.a.p

sage: E.q_eigenform(10)

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

## 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.