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Properties

Label 412224.b
Number of curves $2$
Conductor $412224$
CM no
Rank $0$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 412224.b have rank \(0\).

Complex multiplication

The elliptic curves in class 412224.b do not have complex multiplication.

Modular form 412224.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{5} + 4 q^{7} + q^{9} + 4 q^{15} - 6 q^{17} + q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 412224.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412224.b1 412224b1 \([0, -1, 0, -31672545, 40482638721]\) \(13403946614821979039929/5057590268826067968\) \(1325816943431140761403392\) \([2]\) \(121405440\) \(3.3282\) \(\Gamma_0(N)\)-optimal
412224.b2 412224b2 \([0, -1, 0, 99082015, 288785548161]\) \(410363075617640914325831/374944243169850027552\) \(-98289383681517165622591488\) \([2]\) \(242810880\) \(3.6748\)