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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 248430ge
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.ge7 | 248430ge1 | \([1, 1, 1, -28350176, -85819090687]\) | \(-4437543642183289/3033210136320\) | \(-1722466771395218048229120\) | \([2]\) | \(55738368\) | \(3.3509\) | \(\Gamma_0(N)\)-optimal |
| 248430.ge6 | 248430ge2 | \([1, 1, 1, -511298096, -4449350137471]\) | \(26031421522845051769/5797789779600\) | \(3292386545632406240163600\) | \([2, 2]\) | \(111476736\) | \(3.6975\) | |
| 248430.ge8 | 248430ge3 | \([1, 1, 1, 229892809, 1278398778509]\) | \(2366200373628880151/2612420149248000\) | \(-1483513076170299348615168000\) | \([2]\) | \(167215104\) | \(3.9002\) | |
| 248430.ge5 | 248430ge4 | \([1, 1, 1, -569430716, -3374873295487]\) | \(35958207000163259449/12145729518877500\) | \(6897186337377260714903977500\) | \([2]\) | \(222953472\) | \(4.0441\) | |
| 248430.ge3 | 248430ge5 | \([1, 1, 1, -8180332196, -284780155015231]\) | \(106607603143751752938169/5290068420\) | \(3004067196911114645220\) | \([2]\) | \(222953472\) | \(4.0441\) | |
| 248430.ge4 | 248430ge6 | \([1, 1, 1, -1296461111, 11989129505933]\) | \(424378956393532177129/136231857216000000\) | \(77361882861406228577856000000\) | \([2, 2]\) | \(334430208\) | \(4.2468\) | |
| 248430.ge1 | 248430ge7 | \([1, 1, 1, -18762083831, 988995105715469]\) | \(1286229821345376481036009/247265484375000000\) | \(140414465667586822359375000000\) | \([2]\) | \(668860416\) | \(4.5934\) | |
| 248430.ge2 | 248430ge8 | \([1, 1, 1, -8252501111, -279499553070067]\) | \(109454124781830273937129/3914078300576808000\) | \(2222684716978460337831265128000\) | \([2]\) | \(668860416\) | \(4.5934\) |
Rank
sage: E.rank()
The elliptic curves in class 248430ge have rank \(1\).
Complex multiplication
The elliptic curves in class 248430ge do not have complex multiplication.Modular form 248430.2.a.ge
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.
