Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 211185.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
211185.g1 | 211185l8 | \([1, -1, 1, -422370068, -3340981629768]\) | \(242970740812818720001/24375\) | \(835975901694375\) | \([2]\) | \(21233664\) | \(3.2109\) | |
211185.g2 | 211185l6 | \([1, -1, 1, -26398193, -52197624768]\) | \(59319456301170001/594140625\) | \(20376912603800390625\) | \([2, 2]\) | \(10616832\) | \(2.8644\) | |
211185.g3 | 211185l7 | \([1, -1, 1, -25764638, -54822569844]\) | \(-55150149867714721/5950927734375\) | \(-204095679124603271484375\) | \([2]\) | \(21233664\) | \(3.2109\) | |
211185.g4 | 211185l4 | \([1, -1, 1, -1689548, -773992794]\) | \(15551989015681/1445900625\) | \(49589254512608630625\) | \([2, 2]\) | \(5308416\) | \(2.5178\) | |
211185.g5 | 211185l2 | \([1, -1, 1, -373703, 74464062]\) | \(168288035761/27720225\) | \(950705234442911025\) | \([2, 2]\) | \(2654208\) | \(2.1712\) | |
211185.g6 | 211185l1 | \([1, -1, 1, -357458, 82346136]\) | \(147281603041/5265\) | \(180570794765985\) | \([2]\) | \(1327104\) | \(1.8246\) | \(\Gamma_0(N)\)-optimal |
211185.g7 | 211185l3 | \([1, -1, 1, 682222, 418273242]\) | \(1023887723039/2798036865\) | \(-95962723741229834385\) | \([2]\) | \(5308416\) | \(2.5178\) | |
211185.g8 | 211185l5 | \([1, -1, 1, 1965577, -3667389744]\) | \(24487529386319/183539412225\) | \(-6294749769487178219025\) | \([2]\) | \(10616832\) | \(2.8644\) |
Rank
sage: E.rank()
The elliptic curves in class 211185.g have rank \(1\).
Complex multiplication
The elliptic curves in class 211185.g do not have complex multiplication.Modular form 211185.2.a.g
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.