Properties

 Label 51376.w Number of curves $3$ Conductor $51376$ CM no Rank $1$ Graph

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Show commands for: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

Elliptic curves in class 51376.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51376.w1 51376x3 $$[0, -1, 0, -2080277, 1155555101]$$ $$-50357871050752/19$$ $$-375641583616$$ $$[]$$ $$466560$$ $$2.0091$$
51376.w2 51376x2 $$[0, -1, 0, -25237, 1650141]$$ $$-89915392/6859$$ $$-135606611685376$$ $$[]$$ $$155520$$ $$1.4598$$
51376.w3 51376x1 $$[0, -1, 0, 1803, 701]$$ $$32768/19$$ $$-375641583616$$ $$[]$$ $$51840$$ $$0.91045$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 51376.w have rank $$1$$.

Complex multiplication

The elliptic curves in class 51376.w do not have complex multiplication.

Modular form 51376.2.a.w

sage: E.q_eigenform(10)

$$q + 2q^{3} - 3q^{5} - q^{7} + q^{9} + 3q^{11} - 6q^{15} - 3q^{17} + q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.