Properties

 Label 770g Number of curves $4$ Conductor $770$ CM no Rank $0$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

Elliptic curves in class 770g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
770.g4 770g1 $$[1, 0, 0, 10, 100]$$ $$109902239/4312000$$ $$-4312000$$ $$$$ $$192$$ $$-0.047062$$ $$\Gamma_0(N)$$-optimal
770.g2 770g2 $$[1, 0, 0, -270, 1612]$$ $$2177286259681/105875000$$ $$105875000$$ $$$$ $$384$$ $$0.29951$$
770.g3 770g3 $$[1, 0, 0, -90, -2720]$$ $$-80677568161/3131816380$$ $$-3131816380$$ $$$$ $$576$$ $$0.50224$$
770.g1 770g4 $$[1, 0, 0, -3520, -80238]$$ $$4823468134087681/30382271150$$ $$30382271150$$ $$$$ $$1152$$ $$0.84882$$

Rank

sage: E.rank()

The elliptic curves in class 770g have rank $$0$$.

Complex multiplication

The elliptic curves in class 770g do not have complex multiplication.

Modular form770.2.a.g

sage: E.q_eigenform(10)

$$q + q^{2} - 2 q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} - 2 q^{12} + 2 q^{13} + q^{14} - 2 q^{15} + q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})$$ 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 