Properties

Label 1470.b
Number of curves $8$
Conductor $1470$
CM no
Rank $0$
Graph

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

Elliptic curves in class 1470.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1470.b1 1470b7 \([1, 1, 0, -17210393, 27473941797]\) \(4791901410190533590281/41160000\) \(4842432840000\) \([2]\) \(55296\) \(2.4752\)  
1470.b2 1470b6 \([1, 1, 0, -1075673, 428924133]\) \(1169975873419524361/108425318400\) \(12756130284441600\) \([2, 2]\) \(27648\) \(2.1286\)  
1470.b3 1470b8 \([1, 1, 0, -997273, 494199973]\) \(-932348627918877961/358766164249920\) \(-42208480457838838080\) \([2]\) \(55296\) \(2.4752\)  
1470.b4 1470b4 \([1, 1, 0, -213518, 37218672]\) \(9150443179640281/184570312500\) \(21714512695312500\) \([2]\) \(18432\) \(1.9259\)  
1470.b5 1470b3 \([1, 1, 0, -72153, 5639397]\) \(353108405631241/86318776320\) \(10155317715271680\) \([2]\) \(13824\) \(1.7821\)  
1470.b6 1470b2 \([1, 1, 0, -28298, -973692]\) \(21302308926361/8930250000\) \(1050634982250000\) \([2, 2]\) \(9216\) \(1.5793\)  
1470.b7 1470b1 \([1, 1, 0, -24378, -1474668]\) \(13619385906841/6048000\) \(711541152000\) \([2]\) \(4608\) \(1.2328\) \(\Gamma_0(N)\)-optimal
1470.b8 1470b5 \([1, 1, 0, 94202, -7025192]\) \(785793873833639/637994920500\) \(-75059464401904500\) \([2]\) \(18432\) \(1.9259\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1470.b have rank \(0\).

Complex multiplication

The elliptic curves in class 1470.b do not have complex multiplication.

Modular form 1470.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{13} + q^{15} + q^{16} + 6 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.