# Properties

 Label 912.d Number of curves $2$ Conductor $912$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 912.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
912.d1 912f2 $$[0, -1, 0, -70245, 7189389]$$ $$-9358714467168256/22284891$$ $$-91278913536$$ $$[]$$ $$2400$$ $$1.3446$$
912.d2 912f1 $$[0, -1, 0, 315, 2349]$$ $$841232384/1121931$$ $$-4595429376$$ $$[]$$ $$480$$ $$0.53984$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 912.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 912.d do not have complex multiplication.

## Modular form912.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - 3 q^{7} + q^{9} + 3 q^{11} - 6 q^{13} - q^{15} + 3 q^{17} + q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 