Properties

 Label 3136.o Number of curves $4$ Conductor $3136$ CM $$\Q(\sqrt{-7})$$ Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

Elliptic curves in class 3136.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
3136.o1 3136r4 $$[0, 0, 0, -116620, -15327984]$$ $$16581375$$ $$10578455953408$$ $$$$ $$7168$$ $$1.5601$$   $$-28$$
3136.o2 3136r3 $$[0, 0, 0, -6860, -268912]$$ $$-3375$$ $$-10578455953408$$ $$$$ $$3584$$ $$1.2135$$   $$-7$$
3136.o3 3136r2 $$[0, 0, 0, -2380, 44688]$$ $$16581375$$ $$89915392$$ $$$$ $$1024$$ $$0.58716$$   $$-28$$
3136.o4 3136r1 $$[0, 0, 0, -140, 784]$$ $$-3375$$ $$-89915392$$ $$$$ $$512$$ $$0.24058$$ $$\Gamma_0(N)$$-optimal $$-7$$

Rank

sage: E.rank()

The elliptic curves in class 3136.o have rank $$1$$.

Complex multiplication

Each elliptic curve in class 3136.o has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-7})$$.

Modular form3136.2.a.o

sage: E.q_eigenform(10)

$$q - 3 q^{9} + 4 q^{11} + O(q^{20})$$ 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels. 