# Properties

 Label 369600.s Number of curves $2$ Conductor $369600$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 369600.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.s1 369600s1 $$[0, -1, 0, -256033, -33668063]$$ $$1812647208964/568346625$$ $$581986944000000000$$ $$$$ $$4423680$$ $$2.1130$$ $$\Gamma_0(N)$$-optimal
369600.s2 369600s2 $$[0, -1, 0, 715967, -229040063]$$ $$19818563370478/22511671875$$ $$-46103904000000000000$$ $$$$ $$8847360$$ $$2.4595$$

## Rank

sage: E.rank()

The elliptic curves in class 369600.s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 369600.s do not have complex multiplication.

## Modular form 369600.2.a.s

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 