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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 210210.dn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 210210.dn1 | 210210cg8 | \([1, 1, 1, -9555195560, -358706268131485]\) | \(820076206880893214178646273009/2122496008872985839843750\) | \(249709532947897911071777343750\) | \([2]\) | \(445906944\) | \(4.5166\) | |
| 210210.dn2 | 210210cg5 | \([1, 1, 1, -9549252350, -359175727387933]\) | \(818546927584539194367471866449/14273634375000\) | \(1679278810584375000\) | \([2]\) | \(148635648\) | \(3.9673\) | |
| 210210.dn3 | 210210cg7 | \([1, 1, 1, -8709131180, 311570633540627]\) | \(620954771108295351491118574129/2882378618771462717156250\) | \(339108962119843817210715656250\) | \([2]\) | \(445906944\) | \(4.5166\) | |
| 210210.dn4 | 210210cg6 | \([1, 1, 1, -830849930, -812671959373]\) | \(539142086340577084766074129/309580507925165039062500\) | \(36421837176887741680664062500\) | \([2, 2]\) | \(222953472\) | \(4.1701\) | |
| 210210.dn5 | 210210cg4 | \([1, 1, 1, -606089870, -5429142493405]\) | \(209289070072300727183442769/12893854589717635333800\) | \(1516949098625690079386236200\) | \([2]\) | \(148635648\) | \(3.9673\) | |
| 210210.dn6 | 210210cg2 | \([1, 1, 1, -596828870, -5612295438205]\) | \(199841159336796255944706769/834505270358760000\) | \(98178710552437755240000\) | \([2, 2]\) | \(74317824\) | \(3.6208\) | |
| 210210.dn7 | 210210cg1 | \([1, 1, 1, -36723590, -90553545853]\) | \(-46555485820017544148689/3157693080314572800\) | \(-371499433205929175347200\) | \([2]\) | \(37158912\) | \(3.2742\) | \(\Gamma_0(N)\)-optimal |
| 210210.dn8 | 210210cg3 | \([1, 1, 1, 206814250, -101249397565]\) | \(8315279469612171276463151/4849789796887785750000\) | \(-570572919814051105701750000\) | \([2]\) | \(111476736\) | \(3.8235\) |
Rank
sage: E.rank()
The elliptic curves in class 210210.dn have rank \(0\).
Complex multiplication
The elliptic curves in class 210210.dn do not have complex multiplication.Modular form 210210.2.a.dn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 2 & 12 & 6 & 12 & 4 \\ 3 & 1 & 12 & 6 & 4 & 2 & 4 & 12 \\ 4 & 12 & 1 & 2 & 3 & 6 & 12 & 4 \\ 2 & 6 & 2 & 1 & 6 & 3 & 6 & 2 \\ 12 & 4 & 3 & 6 & 1 & 2 & 4 & 12 \\ 6 & 2 & 6 & 3 & 2 & 1 & 2 & 6 \\ 12 & 4 & 12 & 6 & 4 & 2 & 1 & 3 \\ 4 & 12 & 4 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
