Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-8246030x+9116091722\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-8246030xz^2+9116091722z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-131936475x+583297933750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(309, 81070\right) \) | $1.9102153385107191808535517870$ | $\infty$ |
| \( \left(\frac{6651}{4}, -\frac{6655}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([309:81070:1]\) | $1.9102153385107191808535517870$ | $\infty$ |
| \([13302:-6655:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1235, 649800\right) \) | $1.9102153385107191808535517870$ | $\infty$ |
| \( \left(6650, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(309, 81070\right) \), \( \left(309, -81380\right) \), \( \left(1629, 1210\right) \), \( \left(1629, -2840\right) \)
\([309:81070:1]\), \([309:-81380:1]\), \([1629:1210:1]\), \([1629:-2840:1]\)
\((1235,\pm 649800)\), \((6515,\pm 16200)\)
Invariants
| Conductor: | $N$ | = | \( 81225 \) | = | $3^{2} \cdot 5^{2} \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $824722379940796875$ | = | $3^{10} \cdot 5^{6} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{115714886617}{1539} \) | = | $3^{-4} \cdot 11^{3} \cdot 19^{-1} \cdot 443^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5812359279314991693330836311$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.24500866220282609366943236992$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9811108308373943$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.25337849539411$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9102153385107191808535517870$ |
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| Real period: | $\Omega$ | ≈ | $0.25715103320285965492848735919$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9297107835038539467103426176 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.929710784 \approx L'(E,1) & = \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 0.257151 \cdot 1.910215 \cdot 32}{2^2} \\ & \approx 3.929710784\end{aligned}$$
Modular invariants
Modular form 81225.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2211840 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 524 & 455 \\ 505 & 2274 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1519 & 1360 \\ 1060 & 879 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2274 & 2275 \end{array}\right),\left(\begin{array}{rr} 2273 & 8 \\ 2272 & 9 \end{array}\right),\left(\begin{array}{rr} 1823 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1771 & 1770 \\ 1210 & 1891 \end{array}\right),\left(\begin{array}{rr} 1656 & 65 \\ 1225 & 816 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $8$ | \( 9025 = 5^{2} \cdot 19^{2} \) |
| $5$ | additive | $14$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 81225bd
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 57b3, its twist by $285$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{285}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1688960160000.33 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | add | add | ss | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | ? | - | - | 1,1 | 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 |
An entry ? indicates that the invariants have not yet been computed.
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.