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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 65520.cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
65520.cv1 | 65520du7 | \([0, 0, 0, -326257707, -2267864675494]\) | \(1286229821345376481036009/247265484375000000\) | \(738330780096000000000000\) | \([2]\) | \(15925248\) | \(3.5804\) | |
65520.cv2 | 65520du8 | \([0, 0, 0, -143504427, 640907473754]\) | \(109454124781830273937129/3914078300576808000\) | \(11687375180269539459072000\) | \([4]\) | \(15925248\) | \(3.5804\) | |
65520.cv3 | 65520du5 | \([0, 0, 0, -142249467, 653016395306]\) | \(106607603143751752938169/5290068420\) | \(15796059661025280\) | \([4]\) | \(5308416\) | \(3.0311\) | |
65520.cv4 | 65520du6 | \([0, 0, 0, -22544427, -27493294246]\) | \(424378956393532177129/136231857216000000\) | \(406786145937260544000000\) | \([2, 2]\) | \(7962624\) | \(3.2338\) | |
65520.cv5 | 65520du4 | \([0, 0, 0, -9901947, 7738316714]\) | \(35958207000163259449/12145729518877500\) | \(36266954011695912960000\) | \([2]\) | \(5308416\) | \(3.0311\) | |
65520.cv6 | 65520du2 | \([0, 0, 0, -8891067, 10202235626]\) | \(26031421522845051769/5797789779600\) | \(17312107517249126400\) | \([2, 2]\) | \(2654208\) | \(2.6845\) | |
65520.cv7 | 65520du1 | \([0, 0, 0, -492987, 196763114]\) | \(-4437543642183289/3033210136320\) | \(-9057116935689338880\) | \([2]\) | \(1327104\) | \(2.3379\) | \(\Gamma_0(N)\)-optimal |
65520.cv8 | 65520du3 | \([0, 0, 0, 3997653, -2931253414]\) | \(2366200373628880151/2612420149248000\) | \(-7800644766932140032000\) | \([2]\) | \(3981312\) | \(2.8872\) |
Rank
sage: E.rank()
The elliptic curves in class 65520.cv have rank \(1\).
Complex multiplication
The elliptic curves in class 65520.cv do not have complex multiplication.Modular form 65520.2.a.cv
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.