# Properties

 Label 378.e Number of curves $3$ Conductor $378$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 378.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
378.e1 378a3 $$[1, -1, 1, -9560, -357371]$$ $$-545407363875/14$$ $$-2480058$$ $$[]$$ $$324$$ $$0.74319$$
378.e2 378a2 $$[1, -1, 1, -110, -539]$$ $$-7414875/2744$$ $$-54010152$$ $$$$ $$108$$ $$0.19388$$
378.e3 378a1 $$[1, -1, 1, 10, 5]$$ $$4492125/3584$$ $$-96768$$ $$$$ $$36$$ $$-0.35542$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 378.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 378.e do not have complex multiplication.

## Modular form378.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{7} + q^{8} + 5q^{13} + q^{14} + q^{16} + 3q^{17} + 2q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 