# Properties

 Label 463680cd Number of curves $4$ Conductor $463680$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cd1")

sage: E.isogeny_class()

## Elliptic curves in class 463680cd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.cd3 463680cd1 $$[0, 0, 0, -73223148, 258296738128]$$ $$-227196402372228188089/19338934824115200$$ $$-3695727997558451286835200$$ $$$$ $$70778880$$ $$3.4589$$ $$\Gamma_0(N)$$-optimal*
463680.cd2 463680cd2 $$[0, 0, 0, -1194533868, 15890713747792]$$ $$986396822567235411402169/6336721794060000$$ $$1210966392928925122560000$$ $$$$ $$141557760$$ $$3.8055$$ $$\Gamma_0(N)$$-optimal*
463680.cd4 463680cd3 $$[0, 0, 0, 434109012, 9550408912]$$ $$47342661265381757089751/27397579603968000000$$ $$-5235758997515186208768000000$$ $$$$ $$212336640$$ $$4.0082$$
463680.cd1 463680cd4 $$[0, 0, 0, -1736443308, 76403420368]$$ $$3029968325354577848895529/1753440696000000000000$$ $$335087735245111296000000000000$$ $$$$ $$424673280$$ $$4.3548$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 463680cd1.

## Rank

sage: E.rank()

The elliptic curves in class 463680cd have rank $$0$$.

## Complex multiplication

The elliptic curves in class 463680cd do not have complex multiplication.

## Modular form 463680.2.a.cd

sage: E.q_eigenform(10)

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