Show commands:
SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 176400dw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.ci7 | 176400dw1 | \([0, 0, 0, 37040325, -65382745750]\) | \(1023887723039/928972800\) | \(-5099161188512563200000000\) | \([2]\) | \(28311552\) | \(3.4283\) | \(\Gamma_0(N)\)-optimal |
| 176400.ci6 | 176400dw2 | \([0, 0, 0, -188751675, -585381721750]\) | \(135487869158881/51438240000\) | \(282346132215490560000000000\) | \([2, 2]\) | \(56623104\) | \(3.7748\) | |
| 176400.ci5 | 176400dw3 | \([0, 0, 0, -1331823675, 18290166214250]\) | \(47595748626367201/1215506250000\) | \(6671952391280400000000000000\) | \([2, 2]\) | \(113246208\) | \(4.1214\) | |
| 176400.ci4 | 176400dw4 | \([0, 0, 0, -2658351675, -52740864121750]\) | \(378499465220294881/120530818800\) | \(661597490523511987200000000\) | \([2]\) | \(113246208\) | \(4.1214\) | |
| 176400.ci2 | 176400dw5 | \([0, 0, 0, -21176823675, 1186148571214250]\) | \(191342053882402567201/129708022500\) | \(711971452953966240000000000\) | \([2, 2]\) | \(226492416\) | \(4.4680\) | |
| 176400.ci8 | 176400dw6 | \([0, 0, 0, 224024325, 58466829118250]\) | \(226523624554079/269165039062500\) | \(-1477455443789062500000000000000\) | \([2]\) | \(226492416\) | \(4.4680\) | |
| 176400.ci1 | 176400dw7 | \([0, 0, 0, -338829123675, 75913534493914250]\) | \(783736670177727068275201/360150\) | \(1976874782601600000000\) | \([2]\) | \(452984832\) | \(4.8146\) | |
| 176400.ci3 | 176400dw8 | \([0, 0, 0, -21044523675, 1201700568514250]\) | \(-187778242790732059201/4984939585440150\) | \(-27362491626403183562121600000000\) | \([2]\) | \(452984832\) | \(4.8146\) |
Rank
sage: E.rank()
The elliptic curves in class 176400dw have rank \(0\).
Complex multiplication
The elliptic curves in class 176400dw do not have complex multiplication.Modular form 176400.2.a.dw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
