Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-831578652512442x-9233246369930800990284\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-831578652512442xz^2-9233246369930800990284z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13305258440199075x-590927780980829703577250\)
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(homogenize, minimize) |
comment: Define the curve
sage: E = EllipticCurve([1, -1, 0, -831578652512442, -9233246369930800990284])
gp: E = ellinit([1, -1, 0, -831578652512442, -9233246369930800990284])
magma: E := EllipticCurve([1, -1, 0, -831578652512442, -9233246369930800990284]);
oscar: E = EllipticCurve([1, -1, 0, -831578652512442, -9233246369930800990284])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Mordell-Weil group structure
\(\Z\)
magma: MordellWeilGroup(E);
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{100932380270193181679971450326607094538836696166895716577718987675995171000014317494094185660916398946971501801153162194750835851944990135943381684092396788027891720906219725812958229455536946977487260211653285399773261075002928865788674172246824220901184241116045125670999513253428753422071138387815724864449}{2461919171074343756943297577905641903783329107995618094367903190073985413754808245190057978890458800397969315084561119275291822053868298119935282630618685444482834405108028482834275080030928632054333069121720963095475614088796578242445234204767736694364871520586460905763147332511117489986957319876641}, \frac{617766452549134570663461623969918058272234465346004212805136058793638260073789927143630390673611324035605086835517676489066443593898734390026147486868067938181435489443938240399202272866186185165081468448458486566254529410138182058470537364776813441879649964810291403519809650488465440502131358413897949611740973685821892491895531381352824244599714264938164483973845893424812922088429148965984554981366228041509659733036063954936894229140435267764710803498636202}{3862875270602870722816039700440253909468769045053553654123168914036506045909296639081084174195785407117574251925782793147819469603305655844056077589965021838384051888025202419174055250211889785862729388610071174833976931837661436480490023789941453960891836629341479492727213444700517114733495106723516272076271944410903173724737544949974369182791771096109139777106265937394946982446298398626879412111980162269857842290746224565410915877045541181775439}\right)\)
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| $\hat{h}(P)$ | ≈ | $706.43592197640800080535049776$ |
sage: E.gens()
magma: Generators(E);
gp: E.gen
Integral points
None
comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
| Conductor: | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-25656029254705983781976153641545891840000000 $ | = | $-1 \cdot 2^{81} \cdot 3^{11} \cdot 5^{7} \cdot 7^{9} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{138357846491853121383730987168838623}{55816105091607428996184145920} \) | = | $-1 \cdot 2^{-81} \cdot 3^{-5} \cdot 5^{-1} \cdot 19^{-1} \cdot 322351^{3} \cdot 1604497^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $7.2927471298956899209455615712\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $4.4792894175530999091185447285\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.082511858129502\dots$ | |||
| Szpiro ratio: | $8.858466658766757\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $706.43592197640800080535049776\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.00044451172049262819800323717285\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Tamagawa product: | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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| Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 5.0243047535284668017238569936 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 5.024304754 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.000445 \cdot 706.435922 \cdot 16}{1^2} \approx 5.024304754$
# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular invariants
Modular form 418950.2.a.s
\( q - q^{2} + q^{4} - q^{8} - 5 q^{11} + q^{13} + q^{16} + 6 q^{17} - q^{19} + O(q^{20}) \)
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comment: q-expansion of modular form
sage: E.q_eigenform(20)
\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma: ModularForm(E);
For more coefficients, see the Downloads section to the right.
| Modular degree: | 183866941440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{81}$ | Non-split multiplicative | 1 | 1 | 81 | 81 |
| $3$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
| $5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 0 |
| $19$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
comment: Local data
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
comment: mod p Galois image
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
gens = [[9577, 2, 9577, 3], [1, 1, 15959, 0], [1, 0, 2, 1], [1, 2, 0, 1], [3991, 2, 0, 1], [15959, 2, 15958, 3], [13681, 2, 13681, 3], [5321, 2, 5321, 3], [4201, 2, 4201, 3], [7981, 2, 7981, 3]]
GL(2,Integers(15960)).subgroup(gens)
Gens := [[9577, 2, 9577, 3], [1, 1, 15959, 0], [1, 0, 2, 1], [1, 2, 0, 1], [3991, 2, 0, 1], [15959, 2, 15958, 3], [13681, 2, 13681, 3], [5321, 2, 5321, 3], [4201, 2, 4201, 3], [7981, 2, 7981, 3]];
sub<GL(2,Integers(15960))|Gens>;
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 9577 & 2 \\ 9577 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 15959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15959 & 2 \\ 15958 & 3 \end{array}\right),\left(\begin{array}{rr} 13681 & 2 \\ 13681 & 3 \end{array}\right),\left(\begin{array}{rr} 5321 & 2 \\ 5321 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right),\left(\begin{array}{rr} 7981 & 2 \\ 7981 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$4392005035622400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
Isogenies
gp: ellisomat(E)
This curve has no rational isogenies. Its isogeny class 418950s consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 27930bj1, its twist by $105$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.