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SageMath
E = EllipticCurve("ka1")
E.isogeny_class()
Elliptic curves in class 327600.ka
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327600.ka1 | 327600ka8 | \([0, 0, 0, -8156442675, -283483084436750]\) | \(1286229821345376481036009/247265484375000000\) | \(11536418439000000000000000000\) | \([2]\) | \(382205952\) | \(4.3851\) | |
| 327600.ka2 | 327600ka7 | \([0, 0, 0, -3587610675, 80113434219250]\) | \(109454124781830273937129/3914078300576808000\) | \(182615237191711554048000000000\) | \([2]\) | \(382205952\) | \(4.3851\) | |
| 327600.ka3 | 327600ka4 | \([0, 0, 0, -3556236675, 81627049413250]\) | \(106607603143751752938169/5290068420\) | \(246813432203520000000\) | \([2]\) | \(127401984\) | \(3.8358\) | |
| 327600.ka4 | 327600ka6 | \([0, 0, 0, -563610675, -3436661780750]\) | \(424378956393532177129/136231857216000000\) | \(6356033530269696000000000000\) | \([2, 2]\) | \(191102976\) | \(4.0385\) | |
| 327600.ka5 | 327600ka5 | \([0, 0, 0, -247548675, 967289589250]\) | \(35958207000163259449/12145729518877500\) | \(566671156432748640000000000\) | \([2]\) | \(127401984\) | \(3.8358\) | |
| 327600.ka6 | 327600ka2 | \([0, 0, 0, -222276675, 1275279453250]\) | \(26031421522845051769/5797789779600\) | \(270501679957017600000000\) | \([2, 2]\) | \(63700992\) | \(3.4892\) | |
| 327600.ka7 | 327600ka1 | \([0, 0, 0, -12324675, 24595389250]\) | \(-4437543642183289/3033210136320\) | \(-141517452120145920000000\) | \([2]\) | \(31850496\) | \(3.1426\) | \(\Gamma_0(N)\)-optimal |
| 327600.ka8 | 327600ka3 | \([0, 0, 0, 99941325, -366406676750]\) | \(2366200373628880151/2612420149248000\) | \(-121885074483314688000000000\) | \([2]\) | \(95551488\) | \(3.6920\) |
Rank
sage: E.rank()
The elliptic curves in class 327600.ka have rank \(2\).
Complex multiplication
The elliptic curves in class 327600.ka do not have complex multiplication.Modular form 327600.2.a.ka
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
