# Properties

 Label 42978r Number of curves 2 Conductor 42978 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("42978.u1")

sage: E.isogeny_class()

## Elliptic curves in class 42978r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42978.u1 42978r1 [1, 0, 0, -459769306, 3796241996132]  11063808 $$\Gamma_0(N)$$-optimal
42978.u2 42978r2 [1, 0, 0, 2127889334, -194909741524108] [] 77446656

## Rank

sage: E.rank()

The elliptic curves in class 42978r have rank $$1$$.

## Modular form 42978.2.a.u

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2q^{11} + q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} + q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 