Properties

Label 68400.i
Number of curves $4$
Conductor $68400$
CM no
Rank $2$
Graph

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

Elliptic curves in class 68400.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68400.i1 68400ft4 \([0, 0, 0, -3483075, -2501522750]\) \(100162392144121/23457780\) \(1094446183680000000\) \([2]\) \(2359296\) \(2.4519\)  
68400.i2 68400ft3 \([0, 0, 0, -1611075, 765405250]\) \(9912050027641/311647500\) \(14540225760000000000\) \([2]\) \(2359296\) \(2.4519\)  
68400.i3 68400ft2 \([0, 0, 0, -243075, -29402750]\) \(34043726521/11696400\) \(545707238400000000\) \([2, 2]\) \(1179648\) \(2.1053\)  
68400.i4 68400ft1 \([0, 0, 0, 44925, -3194750]\) \(214921799/218880\) \(-10212065280000000\) \([2]\) \(589824\) \(1.7587\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 68400.i have rank \(2\).

Complex multiplication

The elliptic curves in class 68400.i do not have complex multiplication.

Modular form 68400.2.a.i

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.