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Properties

Label 381150.pb
Number of curves $8$
Conductor $381150$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("pb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 381150.pb have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 381150.pb do not have complex multiplication.

Modular form 381150.2.a.pb

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 381150.pb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
381150.pb1 381150pb7 \([1, -1, 1, -6981239255, 210651480470247]\) \(1864737106103260904761/129177711985836360\) \(2606701208412735009618013125000\) \([2]\) \(637009920\) \(4.5843\)  
381150.pb2 381150pb4 \([1, -1, 1, -6860768630, 218731171677747]\) \(1769857772964702379561/691787250\) \(13959704292774925781250\) \([2]\) \(212336640\) \(4.0350\)  
381150.pb3 381150pb6 \([1, -1, 1, -1378334255, -15739498959753]\) \(14351050585434661561/3001282273281600\) \(60563436279229545579225000000\) \([2, 2]\) \(318504960\) \(4.2377\)  
381150.pb4 381150pb3 \([1, -1, 1, -1299926255, -18038264703753]\) \(12038605770121350841/757333463040\) \(15282373583875083840000000\) \([2]\) \(159252480\) \(3.8911\)  
381150.pb5 381150pb2 \([1, -1, 1, -428862380, 3416678052747]\) \(432288716775559561/270140062500\) \(5451206841600125976562500\) \([2, 2]\) \(106168320\) \(3.6884\)  
381150.pb6 381150pb5 \([1, -1, 1, -348004130, 4744370517747]\) \(-230979395175477481/348191894531250\) \(-7026229357070869445800781250\) \([2]\) \(212336640\) \(4.0350\)  
381150.pb7 381150pb1 \([1, -1, 1, -31921880, 31569468747]\) \(178272935636041/81841914000\) \(1651503288322697906250000\) \([2]\) \(53084160\) \(3.3418\) \(\Gamma_0(N)\)-optimal
381150.pb8 381150pb8 \([1, -1, 1, 2970042745, -95010411669753]\) \(143584693754978072519/276341298967965000\) \(-5576342752015305540829453125000\) \([2]\) \(637009920\) \(4.5843\)