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SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 223440bv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 223440.dt4 | 223440bv1 | \([0, 1, 0, -7922336, 8580132084]\) | \(114113060120923921/124104960\) | \(59804976902307840\) | \([2]\) | \(8847360\) | \(2.5066\) | \(\Gamma_0(N)\)-optimal |
| 223440.dt3 | 223440bv2 | \([0, 1, 0, -7985056, 8437306100]\) | \(116844823575501841/3760263939600\) | \(1812034732974081638400\) | \([2, 2]\) | \(17694720\) | \(2.8531\) | |
| 223440.dt5 | 223440bv3 | \([0, 1, 0, 2442144, 28912156020]\) | \(3342636501165359/751262567039460\) | \(-362026146814465759395840\) | \([2]\) | \(35389440\) | \(3.1997\) | |
| 223440.dt2 | 223440bv4 | \([0, 1, 0, -19415776, -21177403276]\) | \(1679731262160129361/570261564022500\) | \(274803518446317987840000\) | \([2, 2]\) | \(35389440\) | \(3.1997\) | |
| 223440.dt6 | 223440bv5 | \([0, 1, 0, 57000704, -146714396620]\) | \(42502666283088696719/43898058864843750\) | \(-21154048931389449600000000\) | \([2]\) | \(70778880\) | \(3.5463\) | |
| 223440.dt1 | 223440bv6 | \([0, 1, 0, -278723776, -1790798918476]\) | \(4969327007303723277361/1123462695162150\) | \(541385779704347792793600\) | \([2]\) | \(70778880\) | \(3.5463\) |
Rank
sage: E.rank()
The elliptic curves in class 223440bv have rank \(2\).
Complex multiplication
The elliptic curves in class 223440bv do not have complex multiplication.Modular form 223440.2.a.bv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
