# Properties

 Label 6378c Number of curves 2 Conductor 6378 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6378.d1")
sage: E.isogeny_class()

## Elliptic curves in class 6378c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
6378.d1 6378c1 [1, 0, 0, -1144386, 471132612] 7 98784 $$\Gamma_0(N)$$-optimal
6378.d2 6378c2 [1, 0, 0, 7869654, -13542161508] 1 691488

## Rank

sage: E.rank()

The elliptic curves in class 6378c have rank $$1$$.

## Modular form6378.2.a.d

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

## 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. 