# Properties

 Label 378.b Number of curves $3$ Conductor $378$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 378.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
378.b1 378f1 $$[1, -1, 0, -141, 681]$$ $$-11527859979/28$$ $$-756$$ $$$$ $$72$$ $$-0.20736$$ $$\Gamma_0(N)$$-optimal
378.b2 378f2 $$[1, -1, 0, -96, 1088]$$ $$-5000211/21952$$ $$-432081216$$ $$$$ $$216$$ $$0.34195$$
378.b3 378f3 $$[1, -1, 0, 849, -25939]$$ $$381790581/1835008$$ $$-325066162176$$ $$[]$$ $$648$$ $$0.89125$$

## Rank

sage: E.rank()

The elliptic curves in class 378.b have rank $$1$$.

## Complex multiplication

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

## Modular form378.2.a.b

sage: E.q_eigenform(10)

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

## 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. 