Properties

 Label 10304.e Number of curves $2$ Conductor $10304$ CM no Rank $1$ Graph

Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

Elliptic curves in class 10304.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.e1 10304p2 $$[0, 1, 0, -10689, 399007]$$ $$1030541881826/62236321$$ $$8157439066112$$ $$[2]$$ $$15360$$ $$1.2299$$
10304.e2 10304p1 $$[0, 1, 0, -10529, 412351]$$ $$1969910093092/7889$$ $$517013504$$ $$[2]$$ $$7680$$ $$0.88336$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 10304.e have rank $$1$$.

Complex multiplication

The elliptic curves in class 10304.e do not have complex multiplication.

Modular form 10304.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{5} + q^{7} + q^{9} - 2q^{11} - 4q^{13} + 4q^{15} - 2q^{17} + O(q^{20})$$

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.