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Properties

Label 123210.i
Number of curves $2$
Conductor $123210$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 123210.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
123210.i1 123210bc2 \([1, -1, 0, -2530468755, 48405720230325]\) \(511189448451769/7077888000\) \(24811239165735868004302848000\) \([]\) \(177230592\) \(4.2540\)  
123210.i2 123210bc1 \([1, -1, 0, -253363140, -1518909536304]\) \(513108539209/12597120\) \(44158675175345331482267520\) \([]\) \(59076864\) \(3.7047\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 123210.i have rank \(0\).

Complex multiplication

The elliptic curves in class 123210.i do not have complex multiplication.

Modular form 123210.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - 6 q^{11} + q^{13} + q^{14} + q^{16} + 6 q^{17} + 7 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.