Properties

 Label 448.a Number of curves $6$ Conductor $448$ CM no Rank $0$ Graph

Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

Elliptic curves in class 448.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.a1 448f6 $$[0, 1, 0, -174753, 28059871]$$ $$2251439055699625/25088$$ $$6576668672$$ $$[2]$$ $$1152$$ $$1.4528$$
448.a2 448f5 $$[0, 1, 0, -10913, 436447]$$ $$-548347731625/1835008$$ $$-481036337152$$ $$[2]$$ $$576$$ $$1.1062$$
448.a3 448f4 $$[0, 1, 0, -2273, 33439]$$ $$4956477625/941192$$ $$246727835648$$ $$[2]$$ $$384$$ $$0.90352$$
448.a4 448f2 $$[0, 1, 0, -673, -6945]$$ $$128787625/98$$ $$25690112$$ $$[2]$$ $$128$$ $$0.35421$$
448.a5 448f1 $$[0, 1, 0, -33, -161]$$ $$-15625/28$$ $$-7340032$$ $$[2]$$ $$64$$ $$0.0076359$$ $$\Gamma_0(N)$$-optimal
448.a6 448f3 $$[0, 1, 0, 287, 3231]$$ $$9938375/21952$$ $$-5754585088$$ $$[2]$$ $$192$$ $$0.55694$$

Rank

sage: E.rank()

The elliptic curves in class 448.a have rank $$0$$.

Complex multiplication

The elliptic curves in class 448.a do not have complex multiplication.

Modular form448.2.a.a

sage: E.q_eigenform(10)

$$q - 2 q^{3} - q^{7} + q^{9} + 4 q^{13} + 6 q^{17} + 2 q^{19} + O(q^{20})$$

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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.