Minimal Weierstrass equation
sage: E = EllipticCurve([0, 1, 1, 202, -4801])
gp: E = ellinit([0, 1, 1, 202, -4801])
magma: E := EllipticCurve([0, 1, 1, 202, -4801]);
Minimal equation
Minimal equation
Simplified equation
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\(y^2+y=x^3+x^2+202x-4801\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+202xz^2-4801z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+261360x-227121840\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
sage: E.gens()
magma: Generators(E);
| $P$ | = |
\(\left(\frac{61}{4}, \frac{359}{8}\right)\)
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| $\hat{h}(P)$ | ≈ | $1.4740983915716442093680400081$ |
Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 9075 \) | = | $3 \cdot 5^{2} \cdot 11^{2}$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-10762233075 $ | = | $-1 \cdot 3^{5} \cdot 5^{2} \cdot 11^{6} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{20480}{243} \) | = | $2^{12} \cdot 3^{-5} \cdot 5$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.60584930230451245734577985772\dots$ | ||
| Stable Faltings height: | $-0.86133798616702287711865182013\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1.4740983915716442093680400081\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.62963966739075306800684119902\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 10 $ = $ 5\cdot1\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L'(E,1) $ ≈ $ 9.2815082097041413567954090823 $ | ||
Modular invariants
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
\( q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} + q^{9} + 2 q^{12} + q^{13} - 6 q^{14} - 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 8400 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
| $11$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
Galois representations
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $5$ | 5B.4.1 | 5.12.0.1 |
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | split | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,3 | 4 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 0 | - | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 9075n
consists of 2 curves linked by isogenies of
degree 5.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/5\Z\) | 2.0.11.1-5625.8-g2 |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.0.119790000.2 | \(\Z/10\Z\) | Not in database |
| $8$ | 8.2.40525144171875.1 | \(\Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/15\Z\) | Not in database |
| $20$ | 20.4.120780545291490852832794189453125.1 | \(\Z/5\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.