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SageMath
E = EllipticCurve("ja1")
E.isogeny_class()
Elliptic curves in class 177450ja
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177450.cc7 | 177450ja1 | \([1, 1, 0, -14464375, 31264847125]\) | \(-4437543642183289/3033210136320\) | \(-228761343513759420000000\) | \([2]\) | \(27869184\) | \(3.1827\) | \(\Gamma_0(N)\)-optimal |
| 177450.cc6 | 177450ja2 | \([1, 1, 0, -260866375, 1621296953125]\) | \(26031421522845051769/5797789779600\) | \(437262873254395256250000\) | \([2, 2]\) | \(55738368\) | \(3.5292\) | |
| 177450.cc8 | 177450ja3 | \([1, 1, 0, 117292250, -465804987500]\) | \(2366200373628880151/2612420149248000\) | \(-197025829502681088000000000\) | \([2]\) | \(83607552\) | \(3.7320\) | |
| 177450.cc3 | 177450ja4 | \([1, 1, 0, -4173638875, 103779874155625]\) | \(106607603143751752938169/5290068420\) | \(398971091566746562500\) | \([2]\) | \(111476736\) | \(3.8758\) | |
| 177450.cc5 | 177450ja5 | \([1, 1, 0, -290525875, 1229702574625]\) | \(35958207000163259449/12145729518877500\) | \(916017446145056045273437500\) | \([2]\) | \(111476736\) | \(3.8758\) | |
| 177450.cc4 | 177450ja6 | \([1, 1, 0, -661459750, -4369688763500]\) | \(424378956393532177129/136231857216000000\) | \(10274455539014121000000000000\) | \([2, 2]\) | \(167215104\) | \(4.0785\) | |
| 177450.cc2 | 177450ja7 | \([1, 1, 0, -4210459750, 101855430236500]\) | \(109454124781830273937129/3914078300576808000\) | \(295195443248888156963625000000\) | \([2]\) | \(334430208\) | \(4.4251\) | |
| 177450.cc1 | 177450ja8 | \([1, 1, 0, -9572491750, -360427794387500]\) | \(1286229821345376481036009/247265484375000000\) | \(18648488521415771484375000000\) | \([2]\) | \(334430208\) | \(4.4251\) |
Rank
sage: E.rank()
The elliptic curves in class 177450ja have rank \(0\).
Complex multiplication
The elliptic curves in class 177450ja do not have complex multiplication.Modular form 177450.2.a.ja
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
