Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-2622x-38480\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-2622xz^2-38480z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-41952x-2462704\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-32, 112\right)\) | \(\left(58, 67\right)\) |
$\hat{h}(P)$ | ≈ | $0.24656160750761126734907090853$ | $1.9228302337014497959127784821$ |
Integral points
\( \left(-32, 112\right) \), \( \left(-32, -113\right) \), \( \left(-22, 92\right) \), \( \left(-22, -93\right) \), \( \left(58, 67\right) \), \( \left(58, -68\right) \), \( \left(68, 312\right) \), \( \left(68, -313\right) \), \( \left(188, 2472\right) \), \( \left(188, -2473\right) \), \( \left(1318, 47812\right) \), \( \left(1318, -47813\right) \)
Invariants
Conductor: | \( 16245 \) | = | $3^{2} \cdot 5 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $514001953125 $ | = | $3^{6} \cdot 5^{9} \cdot 19^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{7575076864}{1953125} \) | = | $2^{15} \cdot 5^{-9} \cdot 19 \cdot 23^{3}$ | 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: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.95638649140851754252690058655\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.083659482786610713172226603893\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0058640564899433\dots$ | |||
Szpiro ratio: | $3.6334932760335827\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.46511076878588046833368889756\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.67989551134031348640581346648\dots$ | 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: | $ 18 $ = $ 2\cdot3^{2}\cdot1 $ | 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^{(2)}(E,1)/2! $ ≈ $ 5.6920810315241253220751504638 $ | 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
$\displaystyle 5.692081032 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.679896 \cdot 0.465111 \cdot 18}{1^2} \approx 5.692081032$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 18144 | 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 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
$19$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
Galois representations
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 |
---|---|---|
$3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1625 & 18 \\ 558 & 389 \end{array}\right),\left(\begin{array}{rr} 1693 & 18 \\ 1692 & 19 \end{array}\right),\left(\begin{array}{rr} 1027 & 18 \\ 693 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1709 & 1692 \\ 0 & 1519 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 1646 & 1651 \end{array}\right)$.
The torsion field $K:=\Q(E[1710])$ is a degree-$9573811200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1710\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$3$ | additive | $2$ | \( 361 = 19^{2} \) |
$5$ | split multiplicative | $6$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
$19$ | additive | $74$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 16245j
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1805b2, its twist by $-3$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-19}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.3.7220.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.6.260642000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.66854673.1 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.990439600.1 | \(\Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$12$ | 12.0.4469547301936929.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.38588388032760475707449435212371.2 | \(\Z/9\Z\) | not in database |
$18$ | 18.6.19123847893057641914829888000000.1 | \(\Z/6\Z\) | 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 | add | split | ord | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 12,3 | - | 3 | 2 | 2 | 2 | 2 | - | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 0,0 | - | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.