# Properties

 Label 912.b Number of curves $4$ Conductor $912$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 912.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
912.b1 912g3 $$[0, -1, 0, -1624, -24656]$$ $$115714886617/1539$$ $$6303744$$ $$$$ $$384$$ $$0.44814$$
912.b2 912g2 $$[0, -1, 0, -104, -336]$$ $$30664297/3249$$ $$13307904$$ $$[2, 2]$$ $$192$$ $$0.10156$$
912.b3 912g1 $$[0, -1, 0, -24, 48]$$ $$389017/57$$ $$233472$$ $$$$ $$96$$ $$-0.24501$$ $$\Gamma_0(N)$$-optimal
912.b4 912g4 $$[0, -1, 0, 136, -1872]$$ $$67419143/390963$$ $$-1601384448$$ $$$$ $$384$$ $$0.44814$$

## Rank

sage: E.rank()

The elliptic curves in class 912.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 912.b do not have complex multiplication.

## Modular form912.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} + q^{9} + 6 q^{13} + 2 q^{15} - 6 q^{17} + q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 