# Properties

 Label 265200dz Number of curves 8 Conductor 265200 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("265200.dz1")

sage: E.isogeny_class()

## Elliptic curves in class 265200dz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
265200.dz7 265200dz1 [0, 1, 0, -838458408, -8998565524812] [2] 159252480 $$\Gamma_0(N)$$-optimal
265200.dz6 265200dz2 [0, 1, 0, -2222906408, 28373223787188] [2, 2] 318504960
265200.dz5 265200dz3 [0, 1, 0, -10317114408, 400722685291188] [2] 477757440
265200.dz4 265200dz4 [0, 1, 0, -32549306408, 2259971694187188] [2] 637009920
265200.dz8 265200dz5 [0, 1, 0, 5952325592, 188591420523188] [2] 637009920
265200.dz2 265200dz6 [0, 1, 0, -164775002408, 25744481864107188] [2, 2] 955514880
265200.dz1 265200dz7 [0, 1, 0, -2636400002408, 1647649523114107188] [2] 1911029760
265200.dz3 265200dz8 [0, 1, 0, -164476210408, 25842499384539188] [2] 1911029760

## Rank

sage: E.rank()

The elliptic curves in class 265200dz have rank $$1$$.

## Modular form 265200.2.a.dz

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.