Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-11957x+506445\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-11957xz^2+506445z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-191307x+32221190\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(47, 192\right) \) | $0.22249272578323680753687000112$ | $\infty$ |
| \( \left(71, 72\right) \) | $0.80987652739742679609855500258$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([47:192:1]\) | $0.22249272578323680753687000112$ | $\infty$ |
| \([71:72:1]\) | $0.80987652739742679609855500258$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(187, 1728\right) \) | $0.22249272578323680753687000112$ | $\infty$ |
| \( \left(283, 864\right) \) | $0.80987652739742679609855500258$ | $\infty$ |
Integral points
\( \left(-105, 824\right) \), \( \left(-105, -720\right) \), \( \left(-97, 912\right) \), \( \left(-97, -816\right) \), \( \left(-1, 720\right) \), \( \left(-1, -720\right) \), \( \left(3, 684\right) \), \( \left(3, -688\right) \), \( \left(47, 192\right) \), \( \left(47, -240\right) \), \( \left(63, -16\right) \), \( \left(63, -48\right) \), \( \left(65, -6\right) \), \( \left(65, -60\right) \), \( \left(71, 72\right) \), \( \left(71, -144\right) \), \( \left(101, 516\right) \), \( \left(101, -618\right) \), \( \left(107, 612\right) \), \( \left(107, -720\right) \), \( \left(285, 4350\right) \), \( \left(285, -4636\right) \), \( \left(335, 5664\right) \), \( \left(335, -6000\right) \), \( \left(371, 6672\right) \), \( \left(371, -7044\right) \), \( \left(42023, 8593392\right) \), \( \left(42023, -8635416\right) \), \( \left(123551, 43366032\right) \), \( \left(123551, -43489584\right) \)
\([-105:824:1]\), \([-105:-720:1]\), \([-97:912:1]\), \([-97:-816:1]\), \([-1:720:1]\), \([-1:-720:1]\), \([3:684:1]\), \([3:-688:1]\), \([47:192:1]\), \([47:-240:1]\), \([63:-16:1]\), \([63:-48:1]\), \([65:-6:1]\), \([65:-60:1]\), \([71:72:1]\), \([71:-144:1]\), \([101:516:1]\), \([101:-618:1]\), \([107:612:1]\), \([107:-720:1]\), \([285:4350:1]\), \([285:-4636:1]\), \([335:5664:1]\), \([335:-6000:1]\), \([371:6672:1]\), \([371:-7044:1]\), \([42023:8593392:1]\), \([42023:-8635416:1]\), \([123551:43366032:1]\), \([123551:-43489584:1]\)
\((-421,\pm 6176)\), \((-389,\pm 6912)\), \((-5,\pm 5760)\), \((11,\pm 5488)\), \((187,\pm 1728)\), \((251,\pm 128)\), \((259,\pm 216)\), \((283,\pm 864)\), \((403,\pm 4536)\), \((427,\pm 5328)\), \((1139,\pm 35944)\), \((1339,\pm 46656)\), \((1483,\pm 54864)\), \((168091,\pm 68915232)\), \((494203,\pm 347422464)\)
Invariants
| Conductor: | $N$ | = | \( 33282 \) | = | $2 \cdot 3^{2} \cdot 43^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-99379519488$ | = | $-1 \cdot 2^{13} \cdot 3^{8} \cdot 43^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{140246460241}{73728} \) | = | $-1 \cdot 2^{-13} \cdot 3^{-2} \cdot 43 \cdot 1483^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0604989874563996976210365663$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.11567384282658221865539313772$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9991827355469977$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8204667449798193$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.16966433756222654594584518836$ |
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| Real period: | $\Omega$ | ≈ | $1.0506042026053109617469767132$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 52 $ = $ 13\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $9.2690034359063091075115159714 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.269003436 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.050604 \cdot 0.169664 \cdot 52}{1^2} \\ & \approx 9.269003436\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 104832 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $13$ | $I_{13}$ | split multiplicative | -1 | 1 | 13 | 13 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $43$ | $1$ | $II$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $13$ | 13B.4.1 | 13.28.0.1 | $28$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13416 = 2^{3} \cdot 3 \cdot 13 \cdot 43 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 13391 & 26 \\ 13390 & 27 \end{array}\right),\left(\begin{array}{rr} 6709 & 8970 \\ 11193 & 9283 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 4471 & 0 \\ 0 & 13415 \end{array}\right),\left(\begin{array}{rr} 8239 & 8970 \\ 9360 & 7999 \end{array}\right),\left(\begin{array}{rr} 10063 & 2262 \\ 0 & 4903 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10063 & 8970 \\ 1131 & 9283 \end{array}\right)$.
The torsion field $K:=\Q(E[13416])$ is a degree-$19193172590592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13416\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 16641 = 3^{2} \cdot 43^{2} \) |
| $3$ | additive | $8$ | \( 3698 = 2 \cdot 43^{2} \) |
| $13$ | good | $2$ | \( 16641 = 3^{2} \cdot 43^{2} \) |
| $43$ | additive | $338$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 33282bg
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 11094g1, 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 |
|---|---|---|---|
| $3$ | 3.1.14792.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1750426112.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.3969227961.1 | \(\Z/13\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.16392985250365744426112595966296064.1 | \(\Z/26\Z\) | 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 | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | add | ord |
| $\lambda$-invariant(s) | 6 | - | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.