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Properties

Label 277200is
Number of curves $8$
Conductor $277200$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("is1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 277200is have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 277200is do not have complex multiplication.

Modular form 277200.2.a.is

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 277200is

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277200.is7 277200is1 \([0, 0, 0, -4221075, 1517607250]\) \(178272935636041/81841914000\) \(3818416339584000000000\) \([2]\) \(10616832\) \(2.8360\) \(\Gamma_0(N)\)-optimal
277200.is5 277200is2 \([0, 0, 0, -56709075, 164282895250]\) \(432288716775559561/270140062500\) \(12603654756000000000000\) \([2, 2]\) \(21233664\) \(3.1826\)  
277200.is4 277200is3 \([0, 0, 0, -171891075, -867370202750]\) \(12038605770121350841/757333463040\) \(35334150051594240000000\) \([2]\) \(31850496\) \(3.3853\)  
277200.is2 277200is4 \([0, 0, 0, -907209075, 10517419395250]\) \(1769857772964702379561/691787250\) \(32276025936000000000\) \([2]\) \(42467328\) \(3.5291\)  
277200.is6 277200is5 \([0, 0, 0, -46017075, 228124827250]\) \(-230979395175477481/348191894531250\) \(-16245241031250000000000000\) \([2]\) \(42467328\) \(3.5291\)  
277200.is3 277200is6 \([0, 0, 0, -182259075, -756836954750]\) \(14351050585434661561/3001282273281600\) \(140027825742226329600000000\) \([2, 2]\) \(63700992\) \(3.7319\)  
277200.is1 277200is7 \([0, 0, 0, -923139075, 10128912885250]\) \(1864737106103260904761/129177711985836360\) \(6026915330411181212160000000\) \([2]\) \(127401984\) \(4.0785\)  
277200.is8 277200is8 \([0, 0, 0, 392732925, -4568458922750]\) \(143584693754978072519/276341298967965000\) \(-12892979644649375040000000000\) \([2]\) \(127401984\) \(4.0785\)