# Properties

 Label 485100.d Number of curves $2$ Conductor $485100$ CM no Rank $2$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 485100.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.d1 485100d2 $$[0, 0, 0, -5692575, 5218487750]$$ $$59466754384/121275$$ $$41605145297100000000$$ $$$$ $$17694720$$ $$2.6511$$ $$\Gamma_0(N)$$-optimal*
485100.d2 485100d1 $$[0, 0, 0, -235200, 137671625]$$ $$-67108864/343035$$ $$-7355195329308750000$$ $$$$ $$8847360$$ $$2.3045$$ $$\Gamma_0(N)$$-optimal*
*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 485100.d1.

## Rank

sage: E.rank()

The elliptic curves in class 485100.d have rank $$2$$.

## Complex multiplication

The elliptic curves in class 485100.d do not have complex multiplication.

## Modular form 485100.2.a.d

sage: E.q_eigenform(10)

$$q - q^{11} - 6q^{13} - 2q^{17} + O(q^{20})$$ ## 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. 