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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 99330.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99330.bg1 | 99330bg7 | \([1, 0, 1, -109445052993, -13936141154910092]\) | \(144981343445855326468668443402981523721/25152324114754019937600000000\) | \(25152324114754019937600000000\) | \([2]\) | \(424673280\) | \(4.8473\) | |
| 99330.bg2 | 99330bg8 | \([1, 0, 1, -12995189313, 229548737085556]\) | \(242700987864553795766033815014261001/117688532785797178869851871129600\) | \(117688532785797178869851871129600\) | \([4]\) | \(424673280\) | \(4.8473\) | |
| 99330.bg3 | 99330bg5 | \([1, 0, 1, -10696531938, 425805659785156]\) | \(135348263676881498043458563567599001/94865985929121315504750000\) | \(94865985929121315504750000\) | \([12]\) | \(141557760\) | \(4.2980\) | |
| 99330.bg4 | 99330bg6 | \([1, 0, 1, -6861941313, -216306499624844]\) | \(35732607610705316025359129586549001/466066466565993496219484160000\) | \(466066466565993496219484160000\) | \([2, 2]\) | \(212336640\) | \(4.5008\) | |
| 99330.bg5 | 99330bg4 | \([1, 0, 1, -1532943618, -13643973660092]\) | \(398384338462181245438428637773721/150127734623908996582031250000\) | \(150127734623908996582031250000\) | \([6]\) | \(141557760\) | \(4.2980\) | |
| 99330.bg6 | 99330bg2 | \([1, 0, 1, -672781938, 6564320785156]\) | \(33678030186128577471400147599001/874270039107335062500000000\) | \(874270039107335062500000000\) | \([2, 6]\) | \(70778880\) | \(3.9515\) | |
| 99330.bg7 | 99330bg3 | \([1, 0, 1, -67168833, -8900146536332]\) | \(-33514065295328439046114492681/34200409058857329274493337600\) | \(-34200409058857329274493337600\) | \([2]\) | \(106168320\) | \(4.1542\) | |
| 99330.bg8 | 99330bg1 | \([1, 0, 1, 7462542, 329471979268]\) | \(45960438227854760543785319/46920791622800342016000000\) | \(-46920791622800342016000000\) | \([6]\) | \(35389440\) | \(3.6049\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 99330.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 99330.bg do not have complex multiplication.Modular form 99330.2.a.bg
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 4 & 12 \\ 4 & 1 & 3 & 2 & 12 & 6 & 4 & 12 \\ 12 & 3 & 1 & 6 & 4 & 2 & 12 & 4 \\ 2 & 2 & 6 & 1 & 6 & 3 & 2 & 6 \\ 3 & 12 & 4 & 6 & 1 & 2 & 12 & 4 \\ 6 & 6 & 2 & 3 & 2 & 1 & 6 & 2 \\ 4 & 4 & 12 & 2 & 12 & 6 & 1 & 3 \\ 12 & 12 & 4 & 6 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
