Properties

 Label 1078.b Number of curves $2$ Conductor $1078$ CM no Rank $1$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 1078.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1078.b1 1078f2 $$[1, 0, 1, -11492, 473190]$$ $$1426487591593/2156$$ $$253651244$$ $$$$ $$1536$$ $$0.88064$$
1078.b2 1078f1 $$[1, 0, 1, -712, 7494]$$ $$-338608873/13552$$ $$-1594379248$$ $$$$ $$768$$ $$0.53407$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 1078.b have rank $$1$$.

Complex multiplication

The elliptic curves in class 1078.b do not have complex multiplication.

Modular form1078.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + 2 q^{6} - q^{8} + q^{9} + 2 q^{10} + q^{11} - 2 q^{12} + 4 q^{13} + 4 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})$$ 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels. 