Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-44772x-3465084\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-44772xz^2-3465084z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-58025187x-160796584674\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-115, 481\right)\) |
$\hat{h}(P)$ | ≈ | $2.4646622518576995143482623823$ |
Torsion generators
\( \left(-\frac{581}{4}, \frac{581}{8}\right) \)
Integral points
\( \left(-115, 481\right) \), \( \left(-115, -366\right) \), \( \left(455, 8176\right) \), \( \left(455, -8631\right) \)
Invariants
Conductor: | \( 25410 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $612761797461660 $ | = | $2^{2} \cdot 3 \cdot 5 \cdot 7^{8} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{5602762882081}{345888060} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-8} \cdot 17761^{3}$ | 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: | $1.5891314755933445707837827763\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.39018383919415929875281098732\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9866121641692004\dots$ | |||
Szpiro ratio: | $4.312539894563085\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.4646622518576995143482623823\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.32947478346563783451841030841\dots$ | 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);
|
Tamagawa product: | $ 16 $ = $ 2\cdot1\cdot1\cdot2\cdot2^{2} $ | 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: | $2$ | 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) $ ≈ $ 3.2481762469869875525735485241 $ | 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
$\displaystyle 3.248176247 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.329475 \cdot 2.464662 \cdot 16}{2^2} \approx 3.248176247$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 163840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 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 |
---|---|---|
$2$ | 2B | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18480 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 17018 & 4631 \\ 8107 & 12486 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18465 & 16 \\ 18464 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 18382 & 18467 \end{array}\right),\left(\begin{array}{rr} 6733 & 15136 \\ 14124 & 10825 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 18476 & 18477 \end{array}\right),\left(\begin{array}{rr} 3928 & 6721 \\ 1199 & 5050 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2696 & 6721 \\ 9823 & 5050 \end{array}\right),\left(\begin{array}{rr} 5281 & 15136 \\ 3608 & 10209 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 18479 \end{array}\right)$.
The torsion field $K:=\Q(E[18480])$ is a degree-$78479622144000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18480\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 25410.n
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 210.c4, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-165}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | 2.0.11.1-44100.5-f4 |
$4$ | \(\Q(\sqrt{-11}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-11})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-11}, \sqrt{30})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.4.42693156000000.21 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.10929447936000000.79 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.48575324160000.141 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | nonsplit | split | nonsplit | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 5 | 2 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 1 | 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.