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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 324870.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 324870.dd1 | 324870dd7 | \([1, 1, 1, -110667869061, 14170299110903949]\) | \(1274090022584975661628188489514561/14072533302105480763470\) | \(1655619470459407706341482030\) | \([2]\) | \(1207959552\) | \(4.7923\) | |
| 324870.dd2 | 324870dd5 | \([1, 1, 1, -6922258911, 221037854331489]\) | \(311802066473807207098058600161/1033693082103011001480900\) | \(121612957416337141313226404100\) | \([2, 2]\) | \(603979776\) | \(4.4457\) | |
| 324870.dd3 | 324870dd4 | \([1, 1, 1, -6811753131, -216392475356247]\) | \(297106512928238351998640242081/3977028808593750000\) | \(467893462302246093750000\) | \([2]\) | \(301989888\) | \(4.0992\) | |
| 324870.dd4 | 324870dd8 | \([1, 1, 1, -3939856761, 412769330789829]\) | \(-57487943130312093140621093761/592356094985924086700006670\) | \(-69690102218998982876169084718830\) | \([2]\) | \(1207959552\) | \(4.7923\) | |
| 324870.dd5 | 324870dd3 | \([1, 1, 1, -624558411, 86810909289]\) | \(229010110533436633465952161/132501160769452503210000\) | \(15588629063365317550153290000\) | \([2, 2]\) | \(301989888\) | \(4.0992\) | |
| 324870.dd6 | 324870dd2 | \([1, 1, 1, -426108411, -3375030270711]\) | \(72727020009972527154752161/265361167808100000000\) | \(31219476031455156900000000\) | \([2, 2]\) | \(150994944\) | \(3.7526\) | |
| 324870.dd7 | 324870dd1 | \([1, 1, 1, -14602491, -100595364087]\) | \(-2926956820564562516641/35459588343029760000\) | \(-4171785108969108234240000\) | \([2]\) | \(75497472\) | \(3.4060\) | \(\Gamma_0(N)\)-optimal |
| 324870.dd8 | 324870dd6 | \([1, 1, 1, 2497942089, 697572007089]\) | \(14651516183052242700771495839/8480668142378708755560900\) | \(-997742126282712706382984324100\) | \([2]\) | \(603979776\) | \(4.4457\) |
Rank
sage: E.rank()
The elliptic curves in class 324870.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 324870.dd do not have complex multiplication.Modular form 324870.2.a.dd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 16 & 4 & 4 & 8 & 16 & 8 \\ 2 & 1 & 8 & 2 & 2 & 4 & 8 & 4 \\ 16 & 8 & 1 & 16 & 4 & 2 & 4 & 8 \\ 4 & 2 & 16 & 1 & 4 & 8 & 16 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 8 & 2 & 1 & 2 & 4 \\ 16 & 8 & 4 & 16 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
