# Properties

 Label 378.d Number of curves $3$ Conductor $378$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 378.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
378.d1 378b3 $$[1, -1, 0, -1062, 13590]$$ $$-545407363875/14$$ $$-3402$$ $$[3]$$ $$108$$ $$0.19388$$
378.d2 378b1 $$[1, -1, 0, -12, 24]$$ $$-7414875/2744$$ $$-74088$$ $$[3]$$ $$36$$ $$-0.35542$$ $$\Gamma_0(N)$$-optimal
378.d3 378b2 $$[1, -1, 0, 93, -235]$$ $$4492125/3584$$ $$-70543872$$ $$[]$$ $$108$$ $$0.19388$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form378.2.a.d

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.