Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-21298x-1089548\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-21298xz^2-1089548z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-27602883x-50419911618\)
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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 |
|---|---|---|
| \( \left(171, 467\right) \) | $1.1779186439312706966913833409$ | $\infty$ |
| \( \left(-84, 382\right) \) | $1.3801973763790637571692015327$ | $\infty$ |
| \( \left(-\frac{421}{4}, \frac{421}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([171:467:1]\) | $1.1779186439312706966913833409$ | $\infty$ |
| \([-84:382:1]\) | $1.3801973763790637571692015327$ | $\infty$ |
| \([-842:421:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6171, 119340\right) \) | $1.1779186439312706966913833409$ | $\infty$ |
| \( \left(-3009, 73440\right) \) | $1.3801973763790637571692015327$ | $\infty$ |
| \( \left(-3774, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-84, 382\right) \), \( \left(-84, -298\right) \), \( \left(-76, 350\right) \), \( \left(-76, -274\right) \), \( \left(-63, 116\right) \), \( \left(-63, -53\right) \), \( \left(167, 131\right) \), \( \left(167, -298\right) \), \( \left(171, 467\right) \), \( \left(171, -638\right) \), \( \left(444, 8566\right) \), \( \left(444, -9010\right) \), \( \left(1068, 34046\right) \), \( \left(1068, -35114\right) \), \( \left(3317, 189227\right) \), \( \left(3317, -192544\right) \), \( \left(6801, 557387\right) \), \( \left(6801, -564188\right) \)
\([-84:382:1]\), \([-84:-298:1]\), \([-76:350:1]\), \([-76:-274:1]\), \([-63:116:1]\), \([-63:-53:1]\), \([167:131:1]\), \([167:-298:1]\), \([171:467:1]\), \([171:-638:1]\), \([444:8566:1]\), \([444:-9010:1]\), \([1068:34046:1]\), \([1068:-35114:1]\), \([3317:189227:1]\), \([3317:-192544:1]\), \([6801:557387:1]\), \([6801:-564188:1]\)
\((-3009,\pm 73440)\), \((-2721,\pm 67392)\), \((-2253,\pm 18252)\), \((6027,\pm 46332)\), \((6171,\pm 119340)\), \((15999,\pm 1898208)\), \((38463,\pm 7469280)\), \((119427,\pm 41231268)\), \((244851,\pm 121130100)\)
Invariants
| Conductor: | $N$ | = | \( 112710 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $113827740561600$ | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 13^{6} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{217482980991353}{23168683200} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{3} \cdot 11^{6} \cdot 13^{-6} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4315524269334299840229896728$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.72324909091937596396060601833$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0191101255152029$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5686670715571926$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3175975064576210222574020671$ |
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| Real period: | $\Omega$ | ≈ | $0.39792107943370969995620115804$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot1\cdot2\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.2915978643453701296622649310 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.291597864 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.397921 \cdot 1.317598 \cdot 48}{2^2} \\ & \approx 6.291597864\end{aligned}$$
Modular invariants
Modular form 112710.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 700416 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13260 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 8842 & 1 \\ 8839 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7957 & 4 \\ 2654 & 9 \end{array}\right),\left(\begin{array}{rr} 10144 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 3316 & 9949 \\ 9945 & 3316 \end{array}\right),\left(\begin{array}{rr} 13257 & 4 \\ 13256 & 5 \end{array}\right),\left(\begin{array}{rr} 11221 & 4 \\ 9182 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[13260])$ is a degree-$378414468956160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13260\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 51 = 3 \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1445 = 5 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 22542 = 2 \cdot 3 \cdot 13 \cdot 17^{2} \) |
| $13$ | split multiplicative | $14$ | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $82$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 112710.a
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{51}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-238 +2 \sqrt{1105}})\) | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | ss | split | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 7 | 2 | 4 | 2 | 2,2 | 3 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| $\mu$-invariant(s) | 1 | 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.