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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 139230ds
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.b4 | 139230ds1 | \([1, -1, 0, -51879146670, -4548163935264300]\) | \(21182375175311718755156119308023521/57778579333189522080000\) | \(42120584333895161596320000\) | \([2]\) | \(270532608\) | \(4.5759\) | \(\Gamma_0(N)\)-optimal |
| 139230.b3 | 139230ds2 | \([1, -1, 0, -51899710590, -4544377874938044]\) | \(21207574048850823872792738495132641/34982717474287728110306250000\) | \(25502401038755753792413256250000\) | \([2, 2]\) | \(541065216\) | \(4.9225\) | |
| 139230.b2 | 139230ds3 | \([1, -1, 0, -68066145810, -1476146901247200]\) | \(47839833887939781795850621588688161/26393794292755443609008789062500\) | \(19241076039418718390967407226562500\) | \([2, 2]\) | \(1082130432\) | \(5.2691\) | |
| 139230.b5 | 139230ds4 | \([1, -1, 0, -36062298090, -7370301234515544]\) | \(-7114696532582636527413245800532641/28073201392582203302585928742500\) | \(-20465363815192426207585142053282500\) | \([2]\) | \(1082130432\) | \(5.2691\) | |
| 139230.b1 | 139230ds5 | \([1, -1, 0, -659904890580, 205077824010712926]\) | \(43595355616903186726969048523604598081/306218213075771927833557128906250\) | \(223233077332237735390663146972656250\) | \([2]\) | \(2164260864\) | \(5.6157\) | |
| 139230.b6 | 139230ds6 | \([1, -1, 0, 265109635440, -11663529494215950]\) | \(2826654455041045345083039379223811839/1716409549902715755405793403906250\) | \(-1251262561879079785690823391447656250\) | \([2]\) | \(2164260864\) | \(5.6157\) |
Rank
sage: E.rank()
The elliptic curves in class 139230ds have rank \(1\).
Complex multiplication
The elliptic curves in class 139230ds do not have complex multiplication.Modular form 139230.2.a.ds
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
