Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+21122x+537412\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+21122xz^2+537412z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+27374085x+24991371990\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(18, 952\right)\) |
$\hat{h}(P)$ | ≈ | $1.0416128323906361911985494631$ |
Integral points
\( \left(18, 952\right) \), \( \left(18, -970\right) \), \( \left(96, 1810\right) \), \( \left(96, -1906\right) \)
Invariants
Conductor: | \( 48050 \) | = | $2 \cdot 5^{2} \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-727043015475200 $ | = | $-1 \cdot 2^{15} \cdot 5^{2} \cdot 31^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{46969655}{32768} \) | = | $2^{-15} \cdot 5 \cdot 211^{3}$ | 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: | $1.5402001504327464359083794007\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.44503310388217674948966265044\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0629647337743247\dots$ | |||
Szpiro ratio: | $3.848591958772426\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.0416128323906361911985494631\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.32080045667251122524731975877\dots$ | 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: | $ 30 $ = $ ( 3 \cdot 5 )\cdot1\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) $ ≈ $ 10.024496169205919472328671367 $ | 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
$\displaystyle 10.024496169 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.320800 \cdot 1.041613 \cdot 30}{1^2} \approx 10.024496169$
Modular invariants
Modular form 48050.2.a.u
For more coefficients, see the Downloads section to the right.
Modular degree: | 183600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
$5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$31$ | $2$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 2481 & 3410 \\ 1240 & 2977 \end{array}\right),\left(\begin{array}{rr} 1 & 2232 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1241 & 930 \\ 1240 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3162 \\ 930 & 1861 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2790 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1860 & 1 \end{array}\right),\left(\begin{array}{rr} 961 & 2760 \\ 960 & 961 \end{array}\right),\left(\begin{array}{rr} 1 & 1116 \\ 1860 & 1 \end{array}\right),\left(\begin{array}{rr} 1919 & 0 \\ 0 & 3719 \end{array}\right),\left(\begin{array}{rr} 1241 & 2480 \\ 1240 & 2481 \end{array}\right),\left(\begin{array}{rr} 1861 & 930 \\ 2325 & 931 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2760 & 1 \end{array}\right),\left(\begin{array}{rr} 2791 & 930 \\ 3255 & 931 \end{array}\right),\left(\begin{array}{rr} 2326 & 465 \\ 2325 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$82280448000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 48050v
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50b2, its twist by $-31$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{465}) \) | \(\Z/3\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-31}) \) | \(\Z/5\Z\) | Not in database |
$3$ | 3.1.200.1 | \(\Z/2\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-31})\) | \(\Z/15\Z\) | Not in database |
$6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.67867621875.2 | \(\Z/3\Z\) | Not in database |
$6$ | 6.2.160871400000.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.1191640000.4 | \(\Z/10\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$12$ | deg 12 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/30\Z\) | Not in database |
$18$ | 18.6.25202239531552416053043284625000000000000.1 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.81946010771699163104952000000000000000.1 | \(\Z/6\Z\) | Not in database |
$20$ | 20.4.3816691632293169386684894561767578125.1 | \(\Z/5\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ord | add | ord | ord | ord | ord | ord | ord | ss | add | ord | ord | ord | ord |
$\lambda$-invariant(s) | 18 | 3 | - | 1 | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.