# Properties

 Label 185020m Number of curves $4$ Conductor $185020$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 185020m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
185020.o4 185020m1 $$[0, -1, 0, -38125, 2819250]$$ $$643956736/15125$$ $$143947243682000$$ $$$$ $$870912$$ $$1.5022$$ $$\Gamma_0(N)$$-optimal
185020.o3 185020m2 $$[0, -1, 0, -84380, -5303128]$$ $$436334416/171875$$ $$26172226124000000$$ $$$$ $$1741824$$ $$1.8488$$
185020.o2 185020m3 $$[0, -1, 0, -374525, -86982730]$$ $$610462990336/8857805$$ $$84301263789926480$$ $$$$ $$2612736$$ $$2.0515$$
185020.o1 185020m4 $$[0, -1, 0, -5971380, -5614436728]$$ $$154639330142416/33275$$ $$5066942977606400$$ $$$$ $$5225472$$ $$2.3981$$

## Rank

sage: E.rank()

The elliptic curves in class 185020m have rank $$2$$.

## Complex multiplication

The elliptic curves in class 185020m do not have complex multiplication.

## Modular form 185020.2.a.m

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 