Properties

 Label 77315c Number of curves $2$ Conductor $77315$ CM no Rank $1$ Graph

Related objects

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

Elliptic curves in class 77315c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77315.e2 77315c1 $$[1, -1, 0, -85, -600]$$ $$-658503/1225$$ $$-127183175$$ $$[2]$$ $$22656$$ $$0.24504$$ $$\Gamma_0(N)$$-optimal
77315.e1 77315c2 $$[1, -1, 0, -1730, -27249]$$ $$5517084663/4375$$ $$454225625$$ $$[2]$$ $$45312$$ $$0.59161$$

Rank

sage: E.rank()

The elliptic curves in class 77315c have rank $$1$$.

Complex multiplication

The elliptic curves in class 77315c do not have complex multiplication.

Modular form 77315.2.a.c

sage: E.q_eigenform(10)

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.