Properties

Label 21312k
Number of curves $3$
Conductor $21312$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 21312k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21312.bh3 21312k1 \([0, 0, 0, -120, -502]\) \(4096000/37\) \(1726272\) \([]\) \(2880\) \(0.019349\) \(\Gamma_0(N)\)-optimal
21312.bh2 21312k2 \([0, 0, 0, -840, 9074]\) \(1404928000/50653\) \(2363266368\) \([]\) \(8640\) \(0.56866\)  
21312.bh1 21312k3 \([0, 0, 0, -67440, 6741002]\) \(727057727488000/37\) \(1726272\) \([]\) \(25920\) \(1.1180\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21312k have rank \(0\).

Complex multiplication

The elliptic curves in class 21312k do not have complex multiplication.

Modular form 21312.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{7} + 3 q^{11} + 4 q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.