Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-6933x-78752\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-6933xz^2-78752z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8984547x-3647288034\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(92, 207)$ | $1.5873277794840621045708583188$ | $\infty$ |
| $(-77, 38)$ | $0$ | $2$ |
Integral points
\( \left(-77, 38\right) \), \( \left(-13, 102\right) \), \( \left(-13, -90\right) \), \( \left(92, 207\right) \), \( \left(92, -300\right) \), \( \left(1148, 38223\right) \), \( \left(1148, -39372\right) \)
Invariants
| Conductor: | $N$ | = | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $18683226132480$ | = | $2^{12} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{7633736209}{3870720} \) | = | $2^{-12} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2390473668726929717378315816$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.043427311858075396288912139182$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9780810097967987$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.640881611248334$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5873277794840621045708583188$ |
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| Real period: | $\Omega$ | ≈ | $0.55204453030539863489017135972$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot3\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2576537107959426639178591094 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.257653711 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.552045 \cdot 1.587328 \cdot 24}{2^2} \\ & \approx 5.257653711\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 9614 & 1691 \end{array}\right),\left(\begin{array}{rr} 7216 & 10101 \\ 195 & 2146 \end{array}\right),\left(\begin{array}{rr} 2536 & 3783 \\ 7241 & 8594 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 10297 & 8424 \\ 702 & 7567 \end{array}\right),\left(\begin{array}{rr} 352 & 9243 \\ 3237 & 4954 \end{array}\right),\left(\begin{array}{rr} 6969 & 8788 \\ 8060 & 5669 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 17745 = 3 \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 35490.bo
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210.d7, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-91}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-195}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-91}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{13})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{-15})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.89015574375.3 | \(\Z/6\Z\) | 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 |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3471607400625.3 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.20138731658459202403144136071483200000000.2 | \(\Z/18\Z\) | 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 | nonsplit | split | split | nonsplit | ss | add | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 4 | 4 | 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 | 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.