Properties

Label 900b
Number of curves $4$
Conductor $900$
CM \(\Q(\sqrt{-3}) \)
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 900b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
900.g4 900b1 \([0, 0, 0, 0, 125]\) \(0\) \(-6750000\) \([2]\) \(144\) \(-0.010696\) \(\Gamma_0(N)\)-optimal \(-3\)
900.g2 900b2 \([0, 0, 0, -375, 2750]\) \(54000\) \(108000000\) \([2]\) \(288\) \(0.33588\)   \(-12\)
900.g3 900b3 \([0, 0, 0, 0, -3375]\) \(0\) \(-4920750000\) \([2]\) \(432\) \(0.53861\)   \(-3\)
900.g1 900b4 \([0, 0, 0, -3375, -74250]\) \(54000\) \(78732000000\) \([2]\) \(864\) \(0.88518\)   \(-12\)

Rank

sage: E.rank()
 

The elliptic curves in class 900b have rank \(0\).

Complex multiplication

Each elliptic curve in class 900b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 900.2.a.b

sage: E.q_eigenform(10)
 
\(q + 4 q^{7} - 2 q^{13} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.