Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-22183x-1312206\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-22183xz^2-1312206z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-28749600x-60877278000\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(145482, 55490055\right)\) |
$\hat{h}(P)$ | ≈ | $8.2915604815434972173951067232$ |
Integral points
\( \left(145482, 55490055\right) \), \( \left(145482, -55490056\right) \)
Invariants
Conductor: | \( 3025 \) | = | $5^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-36842932671875 $ | = | $-1 \cdot 5^{6} \cdot 11^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -32768 \) | = | $-1 \cdot 2^{15}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $1.3805937682867610096601800955\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2225466425290670856866572546\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0251241218312794\dots$ | |||
Szpiro ratio: | $5.20125001555621\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $8.2915604815434972173951067232\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.19524619150157062888586868962\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 2 $ = $ 1\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.2377912112525936753769633141 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.237791211 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.195246 \cdot 8.291560 \cdot 2}{1^2} \approx 3.237791211$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4752 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$5$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$11$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 3025a
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121b1, its twist by $-55$.
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 |
---|---|---|---|
$3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.2.299475.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.0.99825.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.0.89685275625.4 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$8$ | 8.0.5536128125.1 | \(\Z/5\Z\) | Not in database |
$10$ | 10.0.7368586534375.1 | \(\Z/11\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | 16.4.766217865410400390625.1 | \(\Z/5\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | ord | add | ss | add | ss | ss | ss | ord | ss | ord | ord | ss | ss | ord |
$\lambda$-invariant(s) | ? | 3 | - | 1,1 | - | 1,1 | 1,1 | 1,1 | 1 | 1,1 | 3 | 1 | 1,1 | 1,1 | 1 |
$\mu$-invariant(s) | ? | 0 | - | 0,0 | - | 0,0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 | 0,0 | 0,0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.