Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-28033x+1232688\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-28033xz^2+1232688z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2270700x+905441625\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(19, 841)$ | $1.2702180034920978920612665476$ | $\infty$ |
| $(-68, 1682)$ | $2.0014439172071767183528300179$ | $\infty$ |
| $(48, 0)$ | $0$ | $2$ |
Integral points
\((-177,\pm 825)\), \((-68,\pm 1682)\), \((19,\pm 841)\), \((23,\pm 775)\), \( \left(48, 0\right) \), \((217,\pm 2327)\), \((773,\pm 21025)\), \((889,\pm 26071)\), \((163173,\pm 65913375)\)
Invariants
| Conductor: | $N$ | = | \( 84100 \) | = | $2^{2} \cdot 5^{2} \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $743529151250000$ | = | $2^{4} \cdot 5^{7} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{16384}{5} \) | = | $2^{14} \cdot 5^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5584167756947672297179896022$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1609991557021684076464367877$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9562146657916117$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.733504484459869$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5254823524137242377388555121$ |
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| Real period: | $\Omega$ | ≈ | $0.46910317790193909909944475061$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.1082707835152576163166877639 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.108270784 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.469103 \cdot 2.525482 \cdot 24}{2^2} \\ & \approx 7.108270784\end{aligned}$$
Modular invariants
Modular form 84100.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 290304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $29$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 3457 & 24 \\ 3456 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1741 & 2784 \\ 1392 & 2089 \end{array}\right),\left(\begin{array}{rr} 465 & 2146 \\ 1334 & 755 \end{array}\right),\left(\begin{array}{rr} 1161 & 3364 \\ 580 & 1161 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 839 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 3380 & 3461 \end{array}\right),\left(\begin{array}{rr} 608 & 957 \\ 2523 & 86 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$62860492800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $5$ | additive | $18$ | \( 3364 = 2^{2} \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 100 = 2^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 84100.b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 20.a3, its twist by $145$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.269120.4 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{29})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.32925150000.12 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1810639360000.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.45265984000000.26 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1810639360000.25 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.28101827939827750899705000000000000.1 | \(\Z/18\Z\) | 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 | add | ord | add | ord | ss | ord | ord | ord | ord | add | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 6 | - | 2 | 2,4 | 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.